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If you want to learn about the practical applications of Bayesian networks, you've come to the right place.
You will find countless real-world examples and many hours of tutorial videos in this section.
Most examples include the corresponding raw data in CSV format plus the associated BayesiaLab XBL files.
This allows you to replicate all the steps shown in the examples.
Dr. Lionel Jouffe & Stefan Conrady
In this webinar, you'll learn about a new approach to causality analysis using Bayesian networks. We will showcase the innovative causal analysis features of Hellixia, BayesiaLab's new subject matter assistant. All of these new Hellixia features are part of BayesiaLab 11.2, which is now available.
Whether you're a data scientist, researcher, or analytics professional, understanding the causal direction between variables is critical to fully understanding a given problem domain. However, despite many advances in machine learning in recent years, discovering causalities purely from data remains elusive.
You may be familiar with our "Where is my bag?" example, which received much publicity through Judea Pearl's best-selling book, The Book of Why. This example was about correctly reasoning about the probability of receiving a piece of checked luggage after landing at a destination airport.
The key to dealing with this problem was encoding our limited causal knowledge into a Bayesian network, which then allowed us to perform inference correctly. In other words, we provided our knowledge, and then the computer, i.e., the BayesiaLab software, could reason for us.
Today, we once again use an example from the field of air travel. Now, we are interested in the causes and consequences of flight delays. However, we are not relying on any knowledge we might already have to build a Bayesian network model. Instead, we leverage BayesiaLab's new subject matter assistant, Hellixia, to assemble any knowledge that may be accessible through Large Language Models, such as ChatGPT.
To explore the topic of flight delays, we start by evaluating pairwise relationships between variables that are presumably relevant, such as "Holidays" and "Flight Delays" or "Flight Delays" and "Flight Safety."
Without using any data from which a network could potentially be machine-learned, Hellixia utilizes Large Language Models to investigate the presence of causal relationships between selected pairs of variables. This feature determines whether a causal link exists and provides an estimated measure of causal effect, including both positive and negative impacts.
Establishing causality requires much more than discovering associations between variables. It requires that we identify the correct direction and the nature of the influence between the variables of interest. Hellixia can directly tap into the subject matter expertise contained in Large Language Models and use that knowledge to assign causal structural priors to a set of selected arcs. BayesiaLab's Structural Learning algorithms can then use these priors to machine-learn causal Bayesian Networks.
Once causal directions have been established for specific arcs within a Bayesian network, it becomes important to communicate their meaning to stakeholders. Hellixia can elaborate on the causal mechanisms represented by arcs in the Bayesian network, thereby helping stakeholders understand the underlying causal processes.
In our example about flight delays, Hellixia would add comments to the causal arcs shown in the following network, which includes variables such as Crew Availability, Air Traffic, Flight Scheduling, and Passenger Boarding.
Creating a new Bayesian network of a complex domain exclusively from human expert knowledge can be challenging and time-consuming. In this webinar, we'll demonstrate how Hellixia can substantially simplify and accelerate this process. Given a particular variable of interest as a starting point, Hellixia can automatically create nodes and build a comprehensive and fully specified Bayesian network representing the problem domain. Fully specified means that both the causal network structure and the parameters are obtained by Hellixia. As a result, experts can review and build upon such a Hellixia-generated model instead of having to start from zero.
In our example domain, the starting point is "Delays in Scheduled Flight Departures." Hellixia finds causes, such as "Air Traffic" and "Weather Conditions," as well as consequences, e.g., "Passenger Satisfaction" and "Operational Costs."
In addition to building a causal Bayesian network from scratch, like with the #causal-network-generator function above, we can use Hellixia to create a causal Network from a defined set of variables (e.g., created with Hellxia's Dimension Elicitor).
Here, we provide a set of causes and consequences and let Hellixia determine how they are causally related.
Dr. Lionel Jouffe is co-founder and CEO of France-based Bayesia S.A.S. Lionel holds a Ph.D. in Computer Science from the University of Rennes and has worked in Artificial Intelligence since the early 1990s. While working as a Professor/Researcher at ESIEA, Lionel started exploring the potential of Bayesian networks.
After co-founding Bayesia in 2001, he and his team have been working full-time on the development of BayesiaLab. Since then, BayesiaLab has emerged as the leading software package for knowledge discovery, data mining, and knowledge modeling using Bayesian networks. It enjoys broad acceptance in academic communities, business, and industry.
Dr. Lionel Jouffe, Bayesia S.A.S. & Stefan Conrady, Bayesia USA
In the realm of Bayesian Belief Networks, integrating advanced tools can profoundly enhance the modeling, understanding, and interpretation of complex systems. This presentation introduces Hellixia, BayesiaLab's cutting-edge subject matter assistant powered by ChatGPT, as a game-changing tool in this domain.
Discover how Hellixia assists users in identifying pertinent dimensions/nodes within any problem domain. Beyond identification, we delve into exploiting the Independence of Causal Influence principle. This pivotal concept provides a strategic avenue for model simplification, resulting in expedited model-building phases.
The presentation further explores Hellixia's capability to generate embeddings, enabling the discovery of semantic relationships between nodes and facilitating the seamless creation of semantic networks for rapid domain comprehension.
Finally, we'll explore how Hellixia can assist us in identifying causal relationships between nodes.
To provide a hands-on perspective, attendees will be treated to a live demonstration of these features using BayesiaLab, offering a tangible insight into the transformative potential of Hellixia in Bayesian Belief network modeling.
Dr. Lionel Jouffe is co-founder and CEO of France-based Bayesia S.A.S. Lionel holds a Ph.D. in Computer Science from the University of Rennes and has worked in Artificial Intelligence since the early 1990s. While working as a Professor/Researcher at ESIEA, Lionel started exploring the potential of Bayesian networks.
After co-founding Bayesia in 2001, he and his team have been working full-time on the development of BayesiaLab, which has since emerged as the leading software package for knowledge discovery, data mining, and knowledge modeling using Bayesian networks. BayesiaLab enjoys broad acceptance in academic communities, business, and industry.
For each example, we provide an excerpt from Pearl's book to introduce the problem domain and then offer a solution in the form of a Bayesian network.
We share all networks in BayesiaLab's XBL format and publish them as interactive WebSimulators so you can experiment with the models without installing BayesiaLab.
Dr. Lionel Jouffe, Bayesia S.A.S. & Stefan Conrady, Bayesia USA
In this webinar, Dr. Lionel Jouffe, Bayesia's CEO, explains all the development steps from the initial medical knowledge elicitation to releasing a smartphone app for diagnostic decision support — all within the framework of Bayesian networks and BayesiaLab. In this context, you will learn about Bayesia's new REST API and how it can make inferences with Bayesian networks available to any application.
Over the past 12 months, the growing number of COVID-19 infections has led to an urgent need for the reliable detection of new infections. Given the similarity of their symptoms, the common cold, influenza, and COVID-19 have remained difficult to differentiate.
Bayesian networks are recognized as powerful tools for risk analysis and decision support and have become a popular model for clinical decision support systems (CDSS). Bayesian networks are particularly suitable for healthcare as they can
model complex problems with causal dependencies under high degrees of uncertainty;
combine different sources of information, including empirical data and expert opinion;
have an interpretable graphical structure;
model interventions in diagnostic and prognostic ways.
Based on the success of the COVID-19 WebSimulator implementation, we joined forces with Dr. Jordi Ochando at the Spanish Society for Immunology and Dr. Manisha Brahmachary at the Mount Sinai School of Medicine to further refine our web-based diagnostic tool and turn it into an app for iOS and Android. Numerous medical research institutes collaborated in our efforts, including Hôpital Foch (France), the University of Mons (Belgium), and the Hospital Universitario Donostia (Spain).
You can install the app via the following links:
We are starting this new section to share and explain Bayesian networks inspired by .
In this context, the Bayesia team has been at the forefront of developing diagnostic expert systems for COVID-19 using the Bayesian network framework. In March 2020, we assembled a working group of medical researchers, epidemiologists, and clinicians to develop a Bayesian network for distinguishing COVID-19 from other respiratory infections. Since then, we have maintained a public and continuously updated it with the latest understanding of the evolving pandemic.
We present the as a platform for compiling and encoding knowledge from domain specialists. This workflow aggregates the available medical knowledge into a Bayesian network, which is an updatable expert system. Clinicians and patients can then access the Bayesian network expert system through various interfaces, such as the or the .
The has become a popular platform for publishing BayesiaLab models via a web interface to any audience, from local collaborators to the general public. This way, researchers can provide access to a newly-developed Bayesian network so that anyone with an Internet connection can use that model for simulation.
At the core of the is the , which allows you to access the learning and inference functions of BayesiaLab programmatically. As of the beginning of 2021, we've also made this API available as a RESTful service. Your program code can make an HTTP call to Bayesia's API server to perform inference on a Bayesian network model.
For the first implementation of Bayesia's new in a smartphone app, it was an obvious choice for our team to create a COVID-19 diagnostic tool, building on our existing research. This new app allows individuals to perform a COVID-19 self-assessment guided by BayesiaLab's Adaptive Questionnaire algorithm. Since its launch in February, the app has already produced tens of thousands of assessments based on self-reported symptoms.
For Android:
For iOS:
"Let me give you an example in which probabilities make all the difference. It echoes the public debate that erupted in Europe when the smallpox vaccine was first introduced. Unexpectedly, data showed that more people died from smallpox inoculations than from smallpox itself. Naturally, some people used this information to argue that inoculation should be banned, when in fact it was saving lives by eradicating smallpox."
We implement this example as a Causal Bayesian network. "Causal" means that the arc directions represent causal relationships between the variables.
In this network, the green node #Dead is a Function Node that calculates the number of children who died within a population of 1 million.
We created a WebSimulator that allows you to experiment with this model and try out different scenarios: https://simulator.bayesialab.com/#!simulator/685025884871
"I can empathize with the parents who might march to the health department with signs saying, 'Vaccines kill!' And the data seem to be on their side; the vaccinations indeed cause more deaths than smallpox itself. But is logic on their side? Should we ban vaccination or take into account the deaths prevented?" (Pearl, p. 44)
We attempt to answer this counterfactual question in BayesiaLab.
To do so, we need to set Vaccinated=False as Hard Evidence, thus simulating a counterfactual world in which no children are vaccinated.
The Bayesian network infers that not vaccinating would cost the lives of 4,000 children, as shown in the green Function Node.
To replicate the same step in the WebSimulator you need to move the slider Vaccinated=False to 100.
"So far I have emphasized only one aspect of Bayesian networks—namely, the diagram and its arrows that preferably point from cause to effect. Indeed, the diagram is like the engine of the Bayesian network. But like any engine, a Bayesian network runs on fuel. The fuel is called a conditional probability table [...]
Let’s look at a concrete example, suggested by Stefan Conrady and Lionel Jouffe of BayesiaLab, Inc. It’s a scenario familiar to all travelers: we can call it “Where Is My Bag?” Suppose you’ve just landed in Zanzibar after making a tight connection in Aachen, and you’re waiting for your suitcase to appear on the carousel. Other passengers have started to get their bags, but you keep waiting… and waiting… and waiting. What are the chances that your suitcase did not actually make the connection from Aachen to Zanzibar? The answer depends, of course, on how long you have been waiting. If the bags have just started to show up on the carousel, perhaps you should be patient and wait a little bit longer. If you’ve been waiting a long time, then things are looking bad."
"This table, though large, should be easy to understand. The first eleven rows say that if your bag didn’t make it onto the plane (bag on plane = false) then, no matter how much time has elapsed, it won’t be on the carousel (carousel = false). That is, P(carousel = false | bag on plane = false) is 100 percent. That is the meaning of the 100s in the first eleven rows. The other eleven rows say that the bags are unloaded from the plane at a steady rate. If your bag is indeed on the plane, there is a 10 percent probability it will be unloaded in the first minute, a 10 percent probability in the second minute, and so forth. For example, after 5 minutes there is a 50 percent probability it has been unloaded, so we see a 50 for P(carousel = true | bag on plane = true, time = 5). After ten minutes, all the bags have been unloaded, so P(carousel = true | bag on plane = true, time = 10) is 100 percent. Thus we see a 100 in the last entry of the table." (Pearl, p. 119)
"The most interesting thing to do with this Bayesian network, as with most Bayesian networks, is to solve the inverse-probability problem: if x minutes have passed and I still haven’t gotten my bag, what is the probability that it was on the plane? Bayes’s rule automates this computation and reveals an interesting pattern. After one minute, there is still a 47 percent chance that it was on the plane. (Remember that our prior assumption was a 50 percent probability.) After five minutes, the probability drops to 33 percent. After ten minutes, of course, it drops to zero." (Pearl, p. 119)
In BayesiaLab, you can automatically generate and plot the "Curve of Abandoning Hope."
First, you need to define Bag on Plane as your Target Node.
Then set Bag on Carousel=False as Hard Evidence.
Finally, select Main Menu > Analysis > Visual > Target > Target's Posterior > Histogram
.
The x-axis represents Elapsed Time, the y-axis the posterior probability of Bag on Plane=True given Bag on Carousel=False and Elapsed Time.
You can download the XBL file and open it with any version of BayesiaLab: BoW_VaccineSmallpox.xbl
You can find the original and complete description of this example in .
We encoded the problem domain as a Bayesian network in XBL format, which you can download here:
The shown below is associated with the node Bag on Carousel and encodes our assumptions regarding the delivery of bags at the airport.
You can experiment with this model in our WebSimulator: and see how the probabilities evolve as a function of time.
"Subjectivity (Ed, i.e., the prior) is sometimes seen as a deficiency of Bayesian inference. Others regard it as a powerful advantage; it permits us to express our personal experience mathematically and combine it with data in a principled and transparent way. Bayes’s rule informs our reasoning in cases where ordinary intuition fails us or where emotion might lead us astray. We will demonstrate this power in a situation familiar to all of us.
Suppose you take a medical test to see if you have a disease, and it comes back positive. How likely is it that you have the disease? For specificity, let’s say the disease is breast cancer, and the test is a mammogram."
We implement this example as a causal Bayesian network, which means the arc between Breast Cancer and Mammogram represents a causal relationship.
You can also experiment with this model via our WebSimulator: https://simulator.bayesialab.com/#!simulator/186824514911
"Suppose a forty-year-old woman gets a mammogram to check for breast cancer, and it comes back positive. The hypothesis, D (for “disease”), is that she has cancer. The evidence, T (for “test”), is the result of the mammogram. How strongly should she believe the hypothesis? Should she have surgery?" (Pearl, p. 105)
We use the probabilities described by Pearl to set the parameters of the Causal Bayesian Network:
For a typical forty-year-old woman, the probability of getting breast cancer in the next year is about one in seven hundred, 0.14%. We use that as our prior;
The sensitivity (true-positive) of a mammogram is 73%;
The specificity (true-negative) of a mammogram is 88%.
Notice the Input component Breast Cancer—Your Prior Estimate in the WebSimulator. This allows you to set your own initial belief that a patient has breast cancer.
Upon setting Mammogram=Positive as Hard Evidence, the probability of Breast Cancer=True increases from 0.14% to 0.86%.
"The conclusion is startling. I think that most forty-year-old women who have a positive mammogram would be astounded to learn that they still have less than a 1 percent chance of having breast cancer. Figure 3.3 might make the reason easier to understand: the tiny number of true positives (i.e., women with breast cancer) is overwhelmed by the number of false positives."(Pearl, p. 106)
"However, the story would be very different if our patient had a gene that put her at high risk for breast cancer—say, a one-in-twenty chance within the next year. [...]
For a woman in this situation, the chances that the test provides lifesaving information are much higher. That is why the task force continued recommending annual mammograms for high-risk women.
This example shows that P(disease | test) is not the same for everyone; it is context-dependent (Ed: it depends on the prior). If you know that you are at high risk for a disease to begin with, Bayes’s rule allows you to factor that information in. Or if you know that you are immune, you need not even bother with the test!" (Pearl, pp. 107–108)
To answer this question with BayesiaLab, you can either modify the model by setting the prior of Breast Cancer to 5% via the Node Editor, or you can set a Probabilistic Evidence via the Monitor.
In the WebSimulator, you would set the Input Breast Cancer—Your Prior Estimate (initial belief) to 5%.
Upon setting Mammogram=Positive, the probability of Breast Cancer=True increases to 24.25%.
To illustrate the impact of the prior (or prevalence), we added a parent node to Breast Cancer for defining such prior. This is what we call a "hyperparameter."
You can now set Mammogram=Positive as Hard Evidence.
With this evidence set, you can use Target Mean Analysis to explore a range of values for the prior, from 0% to 100%: Main Menu > Analysis > Visual > Target > Target's Posterior > Curves > Total Effects
.
You will obtain a plot in which the x-axis represents the prior of Breast Cancer=True, i.e., the hyper-parameter.
The y-axis represents the updated probability of Breast Cancer=True given a positive mammogram result.
"To see how Bayes’s method works, let’s start with a simple example about customers in a teahouse, for whom we have data documenting their preferences. Data, as we know from Chapter 1, are totally oblivious to cause-effect asymmetries and hence should offer us a way to resolve the inverse-probability puzzle."
Upon completing the data import, the two variables, Tea and Scones, are represented as nodes.
Now we manually add an arc from Tea to Scones to represent a relationship between the nodes.
Then, we let BayesiaLab estimate the probabilities of this relationship using Maximum Likelihood Estimation: Main Menu > Learning > Parameter Estimation
.
Note that the arc between Tea and Scones does not have any causal meaning here. It merely represents the association between Tea and Scones.
As a result, we could invert the arc without changing the representation of this non-causal example.
The following screen capture from the WebSimulator illustrates that the proportion of customers who ordered both tea and scones is indeed 1/3, i.e., the Joint Probability equals 1/3, as shown in the Output Panel on the right.
This innocent-looking equation came to be known as “Bayes’s rule.” If we look carefully at what it says, we find that it offers a general solution to the inverse-probability problem." (Pearl, p. 101)
To answer this question, we need to perform probabilistic inference with the WebSimulator by setting Scones to Yes.
Then, the WebSimulator automatically infers the probability of Tea=Yes, which is now 80%.
"We can also look at Bayes’s rule as a way to update our belief in a particular hypothesis. This is extremely important to understand because a large part of human belief about future events rests on the frequency with which they or similar events have occurred in the past. [...]
As we saw, Bayes’s rule is formally an elementary consequence of his definition of conditional probability. But epistemologically, it is far from elementary. It acts, in fact, as a normative rule for updating beliefs in response to evidence. In other words, we should view Bayes’s rule not just as a convenient definition of the new concept of “conditional probability” but as an empirical claim to faithfully represent the English expression “given that I know.” (Pearl, pp. 101-102)
The resulting Bayesian network in XBL format is available here: BoW_BreastCancer.xbl
The updated network, including the hyperparameter, is available here: BoW_BreastCancer_Prevalence.xbl
To reason about this domain, we first import a small CSV file, which represents Table 3.1 from the book, into BayesiaLab.
The resulting Bayesian network is available in XBL format here:
You can experiment with this model in BayesiaLab or via this WebSimulator page:
"... let denote the probability that a customer orders tea and denote the probability he orders scones. If we already know a customer has ordered tea, then denotes the probability that he orders scones. (Remember that the vertical line stands for “given that.”) Likewise, denotes the probability that he orders tea, given that we already know he ordered scones ...
The Joint Probability of 41.67% corresponds to the Marginal Likelihood , i.e., the prior probability of a customer ordering a scone.
Causal effect estimation is the topic of Chapter 10 in our book, Bayesian Networks & BayesiaLab. In this context, we discuss the central role of confounders and non-confounders in identifying and estimating causal effects. Much of what we explain in that chapter is a practical illustration of Judea Pearl's teaching on causality.
As the originator of an entire school of thought on causality, Judea Pearl is certainly at liberty to take a more light-hearted and playful approach in presenting this serious topic. Chapter 4 in The Book of Why he titled "Confounding and Deconfounding: Or, Slaying the Lurking Variable." In fact, Pearl presents the task of "deconfounding" for causal effect estimation as a series of "games," which we now wish to illustrate with Bayesian networks.
We begin with a selection of quotes from the beginning of Chapter 4 to provide motivation for the forthcoming examples.
"To understand the back-door criterion, it helps first to have an intuitive sense of how information flows in a causal diagram. I like to think of the links as pipes that convey information from a starting point X to a finish Y. Keep in mind that the conveying of information goes in both directions, causal and noncausal, as we saw in Chapter 3.
In fact, the noncausal paths are precisely the source of confounding."
"To deconfound two variables X and Y, we need only to block every noncausal path between them without blocking or perturbing any causal paths."
"With these rules, decounfounding becomes so simple and fun that you can treat it like a game"
For each of the proposed games in Chapter 4, we prepare a corresponding Bayesian network in BayesiaLab. These networks allow you to experiment with the "pipes that convey information" as if they were set up in a laboratory, where you can look inside the tubes and measure the flows in pipes:
"In the mid-1960s, Jacob Yerushalmy pointed out that a mother’s smoking during pregnancy seemed to benefit the health of her newborn baby, if the baby happened to be born underweight."
Pearl, Judea. The Book of Why: The New Science of Cause and Effect (p. 183). Basic Books. Kindle Edition.
We implemented this counterintuitive example as a causal Bayesian network, which means the arcs represent causal relationships.
Since the problem's description in the book is purely qualitative, and no data is available, we associated arbitrary probability distributions with the nodes. Although arbitrary, we specified the probabilities so that the network produces the paradoxical behavior described by Pearl.
Alternatively, you can experiment with this model using our WebSimulator: https://simulator.bayesialab.com/#!simulator/115411982911
The birth-weight paradox can be highlighted with two observations:
Babies of smokers have a lower birth weight than babies of non-smokers.
Low-birth-weight babies of smoking mothers have a higher survival rate compared to those of non-smokers.
"Smoking may be harmful in that it contributes to low birth weight, but certain other causes of low birth weight, such as serious or life-threatening genetic abnormalities, are much more harmful. There are two possible explanations for low birth weight in one particular baby: it might have a smoking mother, or it might be affected by one of those other causes." (Pearl, pp. 184–185)
In other words, Low-Birth-Weight is a collider in the structure Smoking Mother → Low-Birth-Weight ← Birth Defect.
By observing Low-Birth-Weight, we open a noncausal ("back-door") path between Smoking Mother and Mortality of Child, which gives rise to the paradox. Please see our discussion of the Back-Door Criterion for more details on noncausal paths.
In BayesiaLab, we can illustrate what happens by highlighting all information paths:
Set Mortality of Child as Target Node.
Set evidence on Low-Birth-Weight.
Select Smoking Mother. Then, run Main Menu > Analysis > Visual > Graph > Influence Paths to Target
.
Now, all influence paths are visible.
If we observe Smoking Mother=False, this explains away Low-Birth Weight=True and reduces the probability of Birth Defect=True;
On the other hand, if we observe Smoking Mother=False, the probability of Birth Defect=True increases, and the probability of Mortality of Child=True increases, too.
Alternatively, we can use the WebSimulator to replicate these two scenarios:
Dr. Lionel Jouffe, Bayesia S.A.S. & Stefan Conrady, Bayesia USA
Compartmental models represent the most common approach for characterizing the development of an epidemic. In an earlier webinar, we introduced a compartmental S-I-R-D model and created a highly-simplified Bayesian network to illustrate the principles. Given its great relevance, we believe the topic warrants a more detailed explanation beyond the initial "toy model."
For the purpose of this BayesiaLab Tech Talk, we present a more comprehensive S-E-I-R-D model. Each letter denotes a compartment (or state) of individuals in a population:
S: number of susceptible
E: number of exposed
I: number of infected
R: number recovered
D: number of dead
Additionally, we further differentiate within the states of exposed and infected to account for contagiousness and disease severity.
In standard models, a set of differential equations describes how individuals move between the compartments/states. In this Tech Talk, we implement the differential equations as probabilistic, temporal relationships between nodes in a Bayesian network.
While we often use fictional values in webinars to emphasize methodology over the subject matter, we take a different approach here: The numerical values and parameters presented in this Tech Talk are derived from current COVID-19 observations in France. As a result, the model attempts to represent the actual pandemic situation in France and forecast the pandemic progression.
You can download this Bayesian network in XBL format and open it with any version of BayesiaLab: BoW_SmokingNewBorns.xbl
"In Games 1 and 2 you didn’t have to do anything, but this time you do. There is one back-door path from X to Y, X←B→Y, which can only be blocked by controlling for B. If B is unobservable, then there is no way of estimating the effect of X on Y without running a randomized controlled experiment. Some (in fact, most) statisticians in this situation would control for A, as a proxy for the unobservable variable B, but this only partially eliminates the confounding bias and introduces a new collider bias." (Pearl, p. 160)
As with the earlier games, we encode Game 3 as a causal Bayesian network graph:
Again, the probabilities are fictitious and irrelevant.
We select Main Menu > Analysis > Visual > Graph > Influence Paths to Target
to analyze the paths from X to Y.
Given the presence of a noncausal path (highlighted in pink), it becomes clear that we need to control for B to block that path.
Here, "fixing the probabilities" of B are a practical way of controlling for that variable. Note that the states and the values of the variable are irrelevant.
Now, after controlling for B, only one causal path remains, highlighted in blue, which allows us to estimate the effect of X and Y.
However, if B were unobservable ("not observable" or "hidden" in BayesiaLab terminology), some statisticians would perhaps propose to control for A as a proxy of B.
Let's try that scenario as well. We are now fixing A while leaving B "open."
The Influence Path Analysis reveals that controlling for proxy A does not achieve our objective.
Not only does it not block the noncausal path X←B→Y, controlling for A introduces an additional noncausal path X→A ←B→Y, i.e., another bias that prevents us from estimating the effect of X on Y.
This phenomenon is known as "collider bias," as it is produced by conditioning on a collider, such as A.
Stefan Conrady, Bayesia USA
These days, our daily movements are largely governed by "social distancing" mandates. Their purpose is self-explanatory, and following them is certainly the prudent thing to do.
Given the economic impact of keeping individuals apart, the question arises about what kind of distancing is most effective. How much worse is it, for instance, to have groups of 100 people versus groups of 10 in one place? What is the risk of traveling by bus compared to carpooling? How should employees be spaced across open-floor offices once work resumes?
This webinar will neither answer these specific questions nor provide policy recommendations. However, we will endeavor to provide a formal framework for reasoning about such questions.
In this context, the distribution of distances between random points within a given space is very important. For example, if 1000 people were evenly distributed in a square hall, what would be the distribution of their distances, i.e., how many would be close together versus far apart? Unfortunately, the mathematical solution to this question is far from trivial. The distribution of distances l within a unit square is as follows:
P(l) = \left\{ {\begin{array}{*{20}{c}} {2l\left( {{l^2} - 4l + \pi } \right)}&{0 \leqslant l \leqslant 1} \\ {2l\left( { - {l^2} - 4{{\tan }^{ - 1}}\left( {\sqrt {{l^2} - 1} } \right) + 4\sqrt {{l^2} - 1} + \pi - 2} \right)}&{1 < l \leqslant \sqrt 2 } \end{array}} \right.
Presumably, the distances between individuals would be proportional to the probability of transmitting an infection. But that's a secondary question for now. First, we need to consider that individuals in groups are typically not evenly spaced. Secondly, environments can have any shape, although rectangles are very common. And, finally, people move about all the time. With that, an algebraic solution for the distribution of real-world distances between individuals seems unattainable.
Bayesian networks can help with a fairly simple and intuitive solution to this problem. A simple Bayesian network with only five nodes can compute the same distributions as the complicated formula above.
In fact, with this network, we can compute the distributions for any arbitrary positioning of individuals in any type of space. For instance, we could take the U.S. population within the geographic bounds of the country and determine the distribution of distances between any two individuals, and all we need is the above network. Also, we can go beyond Euclidean distances and utilize the great-circle distance between points on the Earth's surface.
Our proposed approach will be the backbone of our discussion about "social distancing." It will allow us to quantify the effect of mandating or changing distances at different levels, similar to using signal filters for attenuating certain frequencies within a frequency range.
Beyond the current pandemic, there are presumably many other applications for this approach. We illustrated one of them in a recent webinar on Geographic Optimization with Bayesian Networks and BayesiaLab. One particular application was finding the optimal warehouse location given the distribution of manufacturing sites and end customers.
A key benefit of this methodology is that changing distributions does not require recalculating the distances of thousands of points. Rather, the Bayesian network can instantly perform inference given new inputs, thus allowing to simulate different configurations, such as desk arrangements in a classroom, etc. It opens up applications, such as maximizing the distances between seat assignments of airplane passengers.
Compared to earlier webinars, we will emphasize the technical aspects of implementing the proposed methodologies with BayesiaLab. In this context, we will also cover foundational elements which may already be familiar to current BayesiaLab users. The objective is that you can replicate all examples presented in the webinar independently after the event.
Stefan Conrady has over 20 years of experience in decision analysis, analytics, market research, and product strategy with Mercedes-Benz, BMW Group, Rolls-Royce Motor Cars, and Nissan, which included assignments in North America, Europe, and Asia.
Today, in his role as Managing Partner of Bayesia USA and Bayesia Singapore, he is recognized as a thought leader in applying Bayesian networks for research, analytics, and reasoning.
Recently, Stefan and his colleague Dr. Lionel Jouffe co-authored Bayesian Networks & BayesiaLab — A Practical Introduction for Researchers, which is now available as an e-book.
This simple network model illustrates how BayesiaLab can quickly learn a Bayesian network classifier from a dataset consisting of 7129 genes from 72 tumor samples.
Golub, Todd R., et al. "Molecular classification of cancer: class discovery and class prediction by gene expression monitoring." science 286.5439 (1999): 531-537.
Stefan Conrady, Bayesia USA
With the outbreak of the COVID-19 pandemic, reasoning about diseases has gone mainstream. No longer is it just healthcare professionals that perform differential diagnoses. Newspapers and social media have been publicizing charts that compare symptoms of COVID-19, the "regular" flu, and the common cold so individuals can potentially self-diagnose and reduce the burden on healthcare providers.
While a chart can list symptoms, it is not an "inference engine." Deliberate reasoning still has to happen in the mind of the self-diagnosing individual to reach a conclusion. That turns out to be the difficult part, as humans are ill-equipped to handle probabilistic inference from effect back to the cause, i.e., from symptom to disease.
In this webinar, we present Bayesian networks as a framework for encoding knowledge about diseases and symptoms. Given this knowledge base, we then use BayesiaLab's inference algorithms to update the probabilities of the potential conditions given the observed symptoms. A very similar model, the so-called "Visit Asia" network, was one of the earliest examples that illustrated the reasoning capabilities of Bayesian networks.
Please note that this webinar does not constitute medical advice. Although the example is based on current events, we focus solely on the reasoning process. Thus, all numerical values and probabilities shown in the presentation should be considered fictional.
Dr. Lionel Jouffe, Bayesia S.A.S. & Stefan Conrady, Bayesia USA
This webinar introduces our collaborative knowledge elicitation project for the differential diagnosis of COVID-19 and influenza-like diseases.
We present a comprehensive knowledge elicitation and reasoning framework that is built on the Bayesian network paradigm. You will see the practical steps for eliciting knowledge with the Bayesia Expert Knowledge Elicitation Environment and see the resulting knowledge base in the form of a Bayesian network. This workflow aggregates emerging medical knowledge and produces an evolving expert system that clinicians can use through a public web portal.
We also briefly present the principles of probabilistic inference and the fundamental challenges that humans — including experts — have with reasoning from symptoms back to their potential causes. In this context, we introduce Bayesian networks as a reasoning framework that can help overcome these cognitive limitations and provide normative inference given the available knowledge.
Please note that the COVID-19 WebSimulator is experimental and not meant to provide medical advice to patients. Always consult your healthcare professional regarding any symptoms or health conditions you may have!
This example is based on a dataset that characterizes the transactions of single-family homes in Ames, Iowa, from 2006 to 2010 (De Cock, 2011)
Comprising 2930 entries, the dataset includes a wide array of explanatory variables, including 23 nominal, 23 ordinal, 14 discrete, and 20 continuous variables.
Dean De Cock. Ames, Iowa: Alternative to the Boston Housing Data as an End of Semester Regression Project. Journal of Statistical Education, 19(3), 2011.
This example features an expert-designed Bayesian network for automated situation assessment in command and control systems. This model provides Combat-ID and Threat Assessment decision support in naval anti-air warfare.
The two attached network examples represent derivations of the well-known "Visit Asia" example, which was first presented in Lauritzen and Spiegelhalter (1988).
Lauritzen, S., & Spiegelhalter, D. (1988). Local Computations with Probabilities on Graphical Structures and Their Application to Expert Systems. Journal of the Royal Statistical Society. Series B (Methodological), 50(2), 157-224. Retrieved June 24, 2020, from www.jstor.org/stable/2345762
Example: Wisconsin Breast Cancer Database
Webinar: Diagnostic Decision Support
Stefan Conrady, Bayesia USA
Everybody knows the meaning of "importance," right? "What is important?" is a common question in daily life, and it is presumably the most common question in research. It's all about attempting to understand what matters within the context of a given domain.
Upon entering the world of statistics and analytics, we encounter a myriad of measures all related to importance, e.g., correlation, weight, significance, indirect/direct effect size, temporal/contemporaneous effects, unit effect, standardized effect, Bayes Factor, Mutual Information, KL-Divergence, contribution, elasticity, etc. Additionally, some of these measures should not be used in isolation but instead need to be seen in conjunction with other quantities, such as joint probability, for decision-making purposes. This highlights that "importance" is not at all a narrowly-defined concept but that it instead covers a broad and diverse spectrum of notions.
While none of these measures are tied to Bayesian networks, we employ this framework to explain major and minor differences between these concepts. More specifically, we attempt to develop an intuition for all of the above concepts using machine-learned Bayesian network models. Our objective is to understand in which contexts what measures of importance are most appropriate to use.
Stefan Conrady has over 20 years of experience in decision analysis, analytics, market research, and product strategy with Mercedes-Benz, BMW Group, Rolls-Royce Motor Cars, and Nissan, which included assignments in North America, Europe, and Asia.
Today, in his role as Managing Partner of Bayesia USA and Bayesia Singapore, he is recognized as a thought leader in applying Bayesian networks for research, analytics, and reasoning.
Recently, Stefan and his colleague Dr. Lionel Jouffe co-authored Bayesian Networks & BayesiaLab — A Practical Introduction for Researchers, which is now available as an e-book.
Stefan Conrady, Bayesia USA
Attribution and contribution often appear in a similar context, and both concepts are closely related to causality. In general, attribution identifies the cause of an observed outcome. In the marketing domain, however, attribution has a somewhat unique interpretation and often refers to the origin of a consumer’s journey toward a purchase. In this particular context, observed outcomes are attributed to specific prior touchpoints, such as website visits or ad clicks.
On the other hand, contribution, as the name implies, refers to the confluence of multiple factors or causes with regard to an effect. In the marketing context, multiple advertising campaigns and promotions, beyond just single touchpoints, would contribute to sales, for instance. So, the definition of contribution is reasonably straightforward.
The decomposition and quantification of the contributing causes is the problem. Plus, this challenge is not new, as this quote from the late 19th century suggests: “Half the money I spend on advertising is wasted; the trouble is, I don’t know which half” (no pun intended, but the attribution of this quote is uncertain). In other words, we do not know how promotional activities contribute to the outcome, i.e., sales. Conversely, calculating the contributions means that we proportionally allocate a given outcome to any number of potential causes.
While contribution appears to be rather straightforward in conceptual terms, a mathematical definition is not nearly as obvious.
We propose distinguishing between two types of contributions, which we shall call Type 1 and Type 2 Contributions. Both types rely on computing the difference between factual and counterfactual outcomes corresponding to factual and counterfactual conditions of multiple causes.
A factual outcome is simply an actual observation of an outcome, e.g., sales. Associated with a factual outcome are multiple causes at their observed, factual levels. A counterfactual outcome results from causes being set to hypothetical, counterfactual conditions. This begs the question of how we can calculate a counterfactual outcome. We need to calculate the counterfactual outcome by simulating a counterfactual intervention using a causal model. In our case, we use a Bayesian network, which provides numerous advantages for our purposes.
We introduce an elementary fictional domain with three causes and one outcome as an example. In fact, we make up the “laws of nature” and, thus, have perfect knowledge of this data-generating process (DGP).
From this generated data, we then machine-learn a Bayesian network that approximates the joint probability distribution of the data as if we did not know the DGP. By default, of course, any machine-learned network would be non-causal. However, by utilizing VanderWeele’s Disjunctive Cause Criterion for confounder selection, we can indeed utilize the learned Bayesian network for causal inference. Hence, we can simulate the effect of setting all three causes to counterfactual states. That choice, however, requires making assumptions from expert knowledge.
In this webinar, we perform machine learning with BayesiaLab and use its Likelihood Matching algorithm for causal inference computations. In addition to calculating contributions, we can determine the “baseline level” of the outcome variable and estimate synergies (positive and negative) between multiple causes.
Stefan Conrady has over 20 years of experience in decision analysis, analytics, market research, and product strategy with Mercedes-Benz, BMW Group, Rolls-Royce Motor Cars, and Nissan, which included assignments in North America, Europe, and Asia.
Today, in his role as Managing Partner of Bayesia USA and Bayesia Singapore, he is recognized as a thought leader in applying Bayesian networks for research, analytics, and reasoning.
Recently, Stefan and his colleague Dr. Lionel Jouffe co-authored Bayesian Networks & BayesiaLab — A Practical Introduction for Researchers, which is now available as an e-book.
Stefan Conrady, Bayesia USA
Even though we may not have any actual observations from a domain, we can still speculate and hypothesize about possible rare events, i.e., we can reason on theoretical grounds as to what could possibly go wrong.
The objective of this webinar is to present Bayesian networks as a framework to merge machine-learned knowledge from data with theoretical knowledge from domain experts to produce a joint probability distribution that includes common and rare events simultaneously.
Stefan Conrady has over 20 years of experience in decision analysis, analytics, market research, and product strategy with Mercedes-Benz, BMW Group, Rolls-Royce Motor Cars, and Nissan, which included assignments in North America, Europe, and Asia.
Today, in his role as Managing Partner of Bayesia USA and Bayesia Singapore, he is recognized as a thought leader in applying Bayesian networks for research, analytics, and reasoning.
The case study we present addresses some of the challenges of Modern Portfolio Theory and was inspired by Rebonato & Denev's book, .
Recently, Stefan and his colleague Dr. Lionel Jouffe co-authored , which is now available as an e-book.
Stefan Conrady, Bayesia USA
In this webinar, we develop a WebSimulator that allows decision-makers to experiment with assumptions for business planning purposes, such as sales forecasts, inventory levels, cost estimates, etc. More specifically, we create a Bayesian network model that explicitly accounts for the uncertainty in all assumptions rather than utilizing single-point forecasts. This Bayesian network serves as the inference engine that drives the WebSimulator output.
Given the Bayesian network, we can also use BayesiaLab's Policy Learning function to formally search for optimal decisions in the presence of uncertainty. Importantly, any such "machine-learned solutions" can be easily replicated by stakeholders, who can individually try out various alternative assumptions and policy scenarios in the WebSimulator.
When the original BayesiaLab WebSimulator was introduced in February 2015, it opened Bayesian network models to a broader audience. The BayesiaLab WebSimulator allows you to publish interactive models via the web without having to install any additional software on the end user's side. Any Bayesian network model built with BayesiaLab can be instantly shared privately and securely with clients or publicly with the wider world.
"Version 1" of the WebSimulator turned out to be a huge success, and it rapidly became an integral part of research workflows. With its growing popularity, however, BayesiaLab users have been developing applications that require greater flexibility and a more sophisticated web interface for the end-user.
With the recent release of BayesiaLab 8, we also introduced an updated WebSimulator. Entirely new is the WebSimulator Editor inside BayesiaLab 8, which allows you to design and configure an elaborate web interface with many customizable elements, including bar charts, gauges, etc. You can immediately review its final appearance with a new preview function.
In this webinar, we will demonstrate all the new features of the WebSimulator by taking you through a complete workflow, from model development to publishing the model via the Bayesia WebSimulator Server.
Stefan Conrady has over 20 years of experience in decision analysis, analytics, market research, and product strategy with Mercedes-Benz, BMW Group, Rolls-Royce Motor Cars, and Nissan, which included assignments in North America, Europe, and Asia.
Today, as the Managing Partner of Bayesia USA and Bayesia Singapore, he is recognized as a thought leader in applying Bayesian networks to research, analytics, and reasoning.
Recently, Stefan and his colleague Dr. Lionel Jouffe co-authored Bayesian Networks & BayesiaLab — A Practical Introduction for Researchers, which is now available as an e-book.
Stefan Conrady, Bayesia USA
Bayesian networks have been gaining prominence among scientists over the last decade, and insights generated with this new paradigm can now be found in books and papers that circulate well beyond the academic community. Practitioners and managerial decision-makers see references to Bayesian networks in studies ranging from biostatistics to marketing analytics. Therefore, it is not surprising that the relatively new Bayesian network framework prompts comparisons with more conventional methods, such as Factor Analysis, which remains widely used in many fields of study.
This webinar aims to compare a traditional statistical factor analysis with BayesiaLab's new workflow for Probabilistic Latent Factor Induction using a psychometric example.
Factor Analysis is a statistical method used to describe variability among observed variables in terms of a potentially lower number of unobserved variables called factors. It is possible, for example, that variations in three or four observed variables mainly reflect the variations in a single unobserved variable or in a reduced number of unobserved variables. The observed variables can be seen as manifestations of abstract underlying (and unobserved) dimensions or (latent) factors.
Probabilistic Latent Factor Induction is a workflow within the BayesiaLab software package, which has the same objective as traditional factor analysis, i.e., variable reduction, but works entirely within the framework of Bayesian networks and is based on principles derived from information theory. This approach also takes advantage of recent advances in machine learning, especially BayesiaLab's Unsupervised Learning algorithms.
Given that factor analysis originated in psychometrics, we shall explore a prototypical psychometric dataset, namely the HEXACO Personality Inventory. The HEXACO model of personality conceptualizes human personality in six dimensions. It was proposed as an alternative to the Big Five/FFM (Five-Factor Model):
Honesty-Humility (H)
Emotionality (E)
Extraversion (X)
Agreeableness (versus Anger) (A)
Conscientiousness (C)
Openness to Experience (O)
Our objective is to reexamine the proposed HEXACO factors and present an alternative latent variable structure. For this purpose, we utilize the publicly available HEXACO dataset from the Open Source Psychometrics Project. Finally, we wish to highlight the speed of the factor induction and validation process with BayesiaLab, which helps researchers focus on the substantive interpretation of results and spend less time dealing with statistical minutiae.
This seminar was recorded on December 3, 2018, at the Virginia Tech Applied Research Center in Arlington, Virginia.
In this workshop, we demonstrate how to elicit human knowledge for developing a high-dimensional computational model of an underlying problem domain in the form of a Bayesian network. This type of "Artificial Intelligence" allows us to reason formally—and quantitatively, despite the absence of numerical data—about the given issue.
As our case study topic, we examine a hypothetical geopolitical scenario that has the characteristics of actual events, such as the sinking of the Russian submarine Kursk in 2000 and the search for the Argentinian submarine San Juan in 2018. More specifically, we reason about the (fictional) disappearance of an American attack submarine in the disputed waters of the South China Sea. Our objective is to determine where and when to launch a search and rescue effort, if at all, given the risk of a military conflict with China.
In this day and age of "Big Data," we may be led to believe that truth can only be established from data, especially in the context of a scientific inquiry. This is a misconception. Even without data, humans do possess knowledge, qualitative or quantitative, tacit or explicit, about many aspects of the world. We believe that a useful amount of knowledge exists regarding the conflict under study. Also, there is one particular type of knowledge that data on its own can never yield, and that is causality. For that, we always have to rely on human expertise.
Although there may not be a single expert among our seminar participants who can fully comprehend all the complexities of our case study topic, there may be several individuals who are more or less knowledgeable about different aspects of the conflict. It is our objective to break down the overall problem into numerous simpler questions, which are perhaps more easily "knowable," at least to some.
So, we are not looking for a single authoritative opinion. Rather, we are looking to collect and consolidate the full spectrum of thought, including causal relationships, from the seminar's participants. This is where the idea of the "wisdom of crowds" comes into play. We want attendees to provide their individual and independent assessments of different elements and relationships within the problem domain.
While the objective of collecting multiple opinions is straightforward, there are many technical and practical challenges in terms of implementing such a process. In the early days of the Cold War, the RAND Corporation proposed the so-called Delphi Method, which facilitated expert knowledge elicitation by iteratively querying stakeholders through a series of questionnaires that were distributed and collected by mail. After each round of questioning, the aggregated results were circulated for review and discussion, and all participating experts could further adjust their assessments based on the collective feedback. Needless to say, before the availability of electronic means of communication, such a process was tedious, bordering on the impractical.
Today, we propose an entirely new, web-based approach, the Bayesia Expert Knowledge Elicitation Environment (BEKEE), which has the same objective as the original Delphi Method. In this seminar, we plan to use BEKEE to collect the opinions of the participants in real time from their own devices via a convenient web interface.
We shall see that systematically eliciting and encoding numerous pieces of (admittedly imperfect) knowledge into a Bayesian network can produce a remarkably useful approximation of the underlying domain, which provides us with a common framework for reasoning about policy options and evaluating their consequences.
Furthermore, by using a Bayesian network model, we can preserve all the uncertainty that exists in our collective knowledge and perform inference by consciously taking into account all the uncertainty. Thus, we manage to avoid two common and problematic extremes in reasoning, i.e., (i) suppressing uncertainty by calculating with the fake precision of single-point estimates, or (ii) being overwhelmed by uncertainty and abandoning quantitative reasoning altogether. A Bayesian network, however, can explicitly represent the uncertainty from the diversity of opinions captured via BEKEE.
Based on the newly generated Bayesian network, we can use BayesiaLab to reason probabilistically about the implications of various hypothetical interventions and simulate the outcomes of different policies.
Stefan Conrady has over 20 years of experience in decision analysis, analytics, market research, and product strategy with Mercedes-Benz, BMW Group, Rolls-Royce Motor Cars, and Nissan, which included assignments in North America, Europe, and Asia.
Today, as the Managing Partner of Bayesia USA and Bayesia Singapore, he is recognized as a thought leader in applying Bayesian networks for research, analytics, and reasoning.
Recently, Stefan and his colleague Dr. Lionel Jouffe co-authored Bayesian Networks & BayesiaLab — A Practical Introduction for Researchers, which is now available as an e-book.
Stefan Conrady, Bayesia USA
This half-day seminar presents a complete workflow for developing a Probabilistic Structural Equation Model (PSEM) based on Bayesian networks and utilizing the BayesiaLab software platform. Our objective is to identify key drivers of satisfaction with a PSEM that is machine-learned from consumer survey data. A key challenge in this context is to resolve the conflict between "driver" as a causal concept versus the non-causal nature of non-experimental survey data. Furthermore, we illustrate how quantifying the joint probability of hypothetical scenarios is critical for establishing priorities for improving customer satisfaction.
Structural Equation Modeling is a statistical technique for testing and estimating causal relations using a combination of statistical data and qualitative causal assumptions. Structural Equation Models (SEM) allow both confirmatory and exploratory modeling, meaning they are suited to both theory testing and theory development.
What we call Probabilistic Structural Equation Models (PSEMs) in BayesiaLab are conceptually similar to traditional SEMs. However, PSEMs are based on a Bayesian network structure as opposed to a series of equations. More specifically, PSEMs can be distinguished from SEMs in terms of key characteristics:
All relationships in a PSEM are probabilistic—hence the name, as opposed to having deterministic relationships plus error terms in traditional SEMs.
PSEMs are nonparametric, facilitating the representation of nonlinear relationships and relationships between categorical variables.
The structure of PSEMs is partially or fully machine-learned from data.
Specifying and estimating a traditional SEM requires a high degree of statistical expertise. Additionally, the multitude of manual steps involved can make the entire SEM workflow extremely time-consuming. On the other hand, the PSEM workflow in BayesiaLab is accessible to non-statistician subject matter experts. Perhaps more importantly, it can be faster by several orders of magnitude. Finally, once a PSEM is validated, it can be utilized like any other Bayesian network. This means that the full array of analysis, simulation, and optimization tools is available to leverage the knowledge represented in the PSEM.
In this seminar, we present a prototypical PSEM application: key drivers analysis and product optimization based on consumer survey data. We examine how consumers perceive product attributes and how these perceptions relate to the consumers’ purchase intent for specific products.
Given the inherent uncertainty of survey data, we also wish to identify higher-level variables, i.e., “latent” variables that represent concepts that are not directly measured in the survey. We do so by analyzing the relationships between the so-called “manifest” variables, i.e., variables directly measured in the survey. Including such concepts helps in building more stable and reliable models than what would be possible using manifest variables only.
Our overall objective is to make surveys clearer to interpret by researchers and make them “actionable” for managerial decision-makers. The ultimate goal is to use the generated PSEM for prioritizing marketing and product initiatives to maximize purchase intent.
Stefan Conrady has over 20 years of experience in decision analysis, analytics, market research, and product strategy with Mercedes-Benz, BMW Group, Rolls-Royce Motor Cars, and Nissan, which included assignments in North America, Europe, and Asia.
Today, in his role as Managing Partner of Bayesia USA and Bayesia Singapore, he is recognized as a thought leader in applying Bayesian networks for research, analytics, and reasoning.
Recently, Stefan and his colleague Dr. Lionel Jouffe co-authored , which is now available as an e-book.
This seminar illustrates how Bayesian networks can serve as a powerful modeling and reasoning framework for health economics research and public policy development.
For five different case studies, we present a complete analysis workflow using the BayesiaLab 8 software platform:
Diagnostic decision support: using a machine-learned Bayesian network for cost-effective evidence-seeking in diagnosing coronary heart disease. This example introduces information-theoretic measures, such as Entropy and Mutual Information.
Quantifying the value of information in field triage for optimizing trauma activation thresholds with regard to hospital resource utilization.
Developing universal health policies under extreme uncertainty, i.e., without any data: "test & treat" or presumptive malaria treatment in sub-Saharan Africa.
Childhood Literacy Campaign: Simpson's Paradox rears its ugly head and leads to misguided policies.
Causal inference from observational healthcare data: using machine learning and the Disjunctive Cause Criterion to reduce—but not eliminate—the need for causal assumptions.
We present the motivation, proposed methodology, and practical implementation for each example.
"Suppose that a prisoner is about to be executed by a firing squad. A certain chain of events must occur for this to happen. First, the court orders the execution. The order goes to a captain, who signals the soldiers on the firing squad (A and B) to fire. We’ll assume that they are obedient and expert marksmen, so they only fire on command, and if either one of them shoots, the prisoner dies."
We implemented Pearl's Firing Squad problem as a Causal Bayesian Network in BayesiaLab.
"Causal" means that the arc directions represent causal relationships between the nodes.
You can download this XBL file and open it with any version of BayesiaLab: BoW_Firing Squad.xbl
Alternatively, you can experiment with the Firing Squad model on this WebSimulator page: https://simulator.bayesialab.com/#!simulator/211001563973
"Using this diagram, we can start answering causal questions from different rungs of the ladder. First, we can answer questions of association (i.e., what one fact tells us about another). If the prisoner is dead, does that mean the court order was given?" (Pearl, p. 40)
To answer this question in BayesiaLab, you set a Hard Positive Evidence on Death=True (double-click on the state True) to indicate that you learned that the prisoner had been executed (first rung of the ladder).
In the WebSimulator, you move the slider Death=True to 100%. The Observed Box is automatically checked upon releasing the mouse button, and the evidence is propagated in the network to update the probability distributions of the other variables.
Once we know that the prisoner is dead, we infer that both soldiers fired and that the court order was given.
"Suppose we find out that A fired. What does that tell us about B? By following the arrows, the computer concludes that B must have fired too. (A would not have fired if the captain hadn’t signaled, so B must have fired as well.) This is true even though A does not cause B (there is no arrow from A to B)." (Pearl, p. 40)
This is another observational query, the "first rung of the ladder."
In BayesiaLab, you set a Hard Positive Evidence on Soldier A=True (double-click on the state True) to indicate that you found out that Soldier A fired.
In the WebSimulator, you move the slider Soldier A=True to 100%.
Given that we know that Soldier A fired, we infer that the court order was given and that Soldier B also fired.
"Going up the Ladder of Causation, we can ask questions about intervention. What if Soldier A decides on his own initiative to fire, without waiting for the captain’s command? Will the prisoner be dead or alive?" (Pearl, p. 40)
You can answer this causal question in BayesiaLab by setting Soldier A to Intervention Mode (Monitor Context Menu > Intervention) and then setting Soldier A=True.
This triggers the "mutilation of the graph" (or "graph surgery"), which blocks the associational path between Soldier A and Death that goes via Captain.
Alternatively, instead of setting Soldier A to Intervention Mode, you can hold constant the probability distribution of Captain to block the path: Monitor Context Menu > Fix Probabilities.
Then, you set Hard Evidence on Soldier A=True.
In the WebSimulator, you can simulate this intervention by first controlling for Court Order, i.e., checking the box Observed, and then setting Soldier A=True to 100%.
If Soldier A decides to fire on his own initiative, this implies the death of the prisoner without affecting our belief regarding the other variables.
The causal Bayesian network for Game 1 is available for download here:
In the spirit of games, we took advantage of BayesiaLab's ability to embellish nodes and added "start" and "finish" icons for the variables X and Y, respectively.
In each game, you need to determine the set of variables you need to adjust for (if any), to estimate the causal effect of X (Start) on Y (Finish) without bias.
As with all networks presented here, we will reason purely based on the causal structure and do not need to consider parameters or numerical values.
For demonstration purposes, the Bayesian networks you can download do contain states and numerical values. However, we chose them arbitrarily, and you should feel free to replace them with any other values of your choice. As long as you maintain the causal structure, the content of the nodes, e.g., numerical or categorical, does not matter at all.
BayesiaLab offers the Influence Paths to Target function, which highlights causal and noncausal paths in a network.
This feature analyzes the paths from the selected node X to the Target Node Y.
To start the function, select Main Menu > Analysis > Visual > Graph > Influence Paths to Target
.
This analysis highlights causal paths in blue and noncausal paths in pink.
However, no pink paths appear, which means that no noncausal paths exist from X to Y.
As a result, no noncausal paths need to be blocked, and, therefore, we do not need to control for any variables to estimate the causal effect of X on Y. The association between X and Y corresponds to the causal effect.
"In this example you should think of A, B, C, and D as “pretreatment” variables. (The treatment, as usual, is X.) Now there is one back-door path X←A→ B←D→E→Y. This path is already blocked by the collider at B, so we don’t need to control for anything." (Pearl, p. 160)
For Game 2, we have once again created a causal Bayesian network, which is available for download here:
Note that the associated probability tables are fictitious. For our purposes, only the causal graph is relevant.
As before, we select Main Menu > Analysis > Visual > Graph > Influence Paths to Target
to analyze the paths from X to Y.
We can see that there is no noncausal path.
Hence, there is then no need to control for any variables.
"Game 5 is just Game 4 with a little extra wrinkle. Now a second back-door path X←B←C→Y needs to be closed. If we close this path by controlling for B, then we open up the M-shaped path X←A→B←C→Y. To close that path, we must control for A or C as well. However, notice that we could just control for C alone; that would close the path X←B←C→Y and not affect the other path." (Pearl, p. 162)
Here we have the causal Bayesian network corresponding to Game 5:
Select Main Menu > Analysis > Visual > Graph > Influence Paths to Target
and see that there is a noncausal path that needs to be blocked.
You can block this path by fixing the probability distribution of variable C.
You can check if this proposed approach is correct by setting your evidence — Fix Probabilities on C — and then running the Influence Paths Analysis again.
"This one introduces a new kind of bias, called 'M-bias' (named for the shape of the graph). [...]
M-bias puts a finger on what is wrong with the traditional approach. It is incorrect to call a variable, like B, a confounder merely because it is associated with both X and Y. To reiterate, X and Y are unconfounded if we do not control for B. B only becomes a confounder when you control for it!" (Pearl, pp. 161–162)
The structure of this example seems simple and can be easily analyzed in BayesiaLab:
Given that B is a collider, there is no open path and, thus, there is no effect of X on Y at all.
As a result, nothing needs to be blocked.
However, as Pearl explains, if one were to apply a traditional three-step for a confounder, one might (incorrectly) conclude that B should be controlled for as a confounder.
Let's try this scenario in BayesiaLab and see what happens.
By controlling for B, we inadvertently open up a noncausal path between X and Y, i.e., we are introducing a bias.
The Influence Path Analysis highlights the M-shape, for which this bias is known.
Judea Pearl concludes Chapter 4 in The Book of Why with a model from a paper by Andrew Forbes and Elizabeth Williamson on the effect of smoking (X) on adult asthma (Y). It is the final example illustrating the Back-Door Criterion. (Pearl, p. 164)
Here is the causal Bayesian network for this problem domain:
You can use Main Menu > Analysis > Visual > Graph > Influence Paths to Target
to find all paths from Smoking to Asthma.
As it turns out, there are 14 noncausal paths and one causal path!
Our task is to block all 14 noncausal paths and keep the one causal path open. If we can't do that, we won't be able to estimate the effect causal effect of Smoking on Asthma.
In this example, the variable Predisposition toward Asthma provides an extra challenge. It is a not-observable (or hidden) variable. Hence, you cannot adjust for it, which means you cannot use it to block any of the 14 noncausal paths.
In the end, you have to adjust for five variables (highlighted in green in the screen capture) to block all noncausal paths to estimate the causal effect of Smoking on Asthma.
After controlling for these variables, only one causal path remains, representing the relationship of interest, i.e., the effect of Smoking on Asthma.
“Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you, ‘Do you want to pick door #2?’ Is it to your advantage to switch your choice of doors?”
Pearl, Judea. The Book of Why: The New Science of Cause and Effect (p. 190). Basic Books. Kindle Edition.
We implemented the Monty Hall Problem as a Causal Bayesian Network in which arcs represent causal relationships.
The game can be described with three nodes, and each one of them with three states: Door #1, Door #2, Door #3.
Your Door, which represents the initial choice that you make as the contestant in this game. We assume a uniform prior distribution, i.e., each door has the same probability of being picked by you.
Location of Car, as the name implies, refers to the door behind which the car is hidden. We also assume a uniform prior distribution, i.e., you do not have any knowledge as to where the prize might be located.
Door Opened is the door that Monty Hall, the game host, opens. He chooses the door according to the following two rules:
He won't open the door that you just selected;
He also knows where the car is, so he won't open the door behind which the car is located.
We represent these two rules in the following Conditional Probability Table:
We can interpret the first row of this Conditional Probability Table as "If you choose Door #1 and the car is behind Door #1, then Monty Hall will open either Door #2 or Door #3."
The second row reads: "If you choose Door #1 and the car is behind Door #2, then Monty Hall can only open Door #3."
Analogously for the third row, "If you choose Door #1 and the car is behind Door #3, then Monty Hall can only open ."
If you experiment with this network in BayesiaLab or the WebSimulator, you will quickly discover that the optimal policy is always to change your door choice.
Let's go through this step by step and try it out in BayesiaLab or the WebSimulator:
You choose Door #1 and set evidence accordingly on the corresponding node: Your Door=Door #1
As per the game rules, Monty Hall cannot open Door #1, which you just picked.
As a result, Monty Hall could only open Door #2 or Door #3.
However, behind one of the two doors is the car, and Monty Hall knows where the car is.
As per the game rules, he won't reveal the car and, therefore, must open a door that presents a goat.
We simulate that Monty Hall opens Door #2 and set the evidence Door Opened=Door #2.
With Door #2 having revealed a goat, the car can only hide behind Door #1 or Door #3.
Given these pieces of evidence set so far, the Bayesian network updates the distribution of the node Location of Car:
Door #1 remains at 1/3.
Door #3 increases from 1/3 to 2/3.
This means that the grand prize, the car, is twice as likely to be behind Door #3 compared to Door #1.
As a result, you should indeed revise your original door selection and pick the other closed door instead.
The optimal decision policy could be different if we also considered the psychological cost of regret and the expected utility from the prizes.
"If I should switch no matter what door I originally chose, then it means that the producers somehow read my mind. How else could they position the car so that it is more likely to be behind the door I did not choose?
The key element in resolving this paradox is that we need to take into account not only the data (i.e., the fact that the host opened a particular door) but also the data-generating process—in other words, the rules of the game." (Pearl, pp. 191–192)
The Causal Bayesian Network we encoded above does indeed represent the data-generating process of this domain: Monty Hall decides which door to open based on two criteria, (1) the choice of the contestant and (2) the location of the car.
This particular arrangement of two causes and their common effect is called a V-structure. In such a V-structure, we call the common effect a Collider. In our example, the node Door Opened is such a Collider.
V-structures have important characteristics: parent nodes, which are initially independent, become dependent if we set any pieces of evidence on their common child node, i.e., the Collider or any of the Collider's descendants. In other words, in a V-structure, parent nodes are marginally independent but conditionally dependent given evidence on their descendants.
Now we know that conditioning on Door Opened allows for information to flow from Your Door to Location of Car, as visualized by the green arrow below:
"It is a bizarre dependence for sure, one of a type that most of us are unaccustomed to. It is a dependence that has no cause." (Pearl, p. 194)
It is precisely the difficulty in understanding this conditional dependency that has made this game so intriguing.
Most casual observers, however, would attempt to reason about this problem the way we illustrate in the following noncausal Bayesian network.
In this context, the following Conditional Probability Tables would apply:
For the node Door Opened: Monty Hall cannot open the door the player chose:
For Location of Car: It cannot be behind the door that Monty Hall opened:
Now that we have formally encoded such a (mis)understanding of the domain, we can simulate the game again:
We pick Door #1, then Monty Hall opens Door #2.
With the given (incorrect) network, we would infer that Door #1 and Door#3 have equal probabilities of containing the car. As a result, there would be no reason to reconsider our initial choice.
The problem that Judea Pearl describes is based on the popular game show, . The show, launched in 1963, was hosted for nearly 30 years by . Given its counterintuitive (and controversial) solution, the stated problem has been debated extensively in academia and popular science and became widely known as the "."
You can download this Bayesian network in XBL format here:
Alternatively, you can experiment with different game scenarios via our WebSimulator:
Please see Structures Within a DAG in of our book to learn more about the important characteristics of different network structures.
The disagreement between the normative choice we explained earlier () and the "common-sense" solution presented just now () has fueled fierce debates and puzzled great minds for decades. With all that has been written about this paradox over the years — and we have used the Monty Hall Problem extensively as an example in our training sessions, we should let the inventor of this puzzle illuminate us. The matter was settled once and for all in 1991 with an experiment at the dining table of Monty Hall's residence in Beverly Hills. New York Times journalist John Tierney shares Monty Hall's perspective on the controversy in his article,
Stefan Conrady, Bayesia USA
The share of renewable energy sources appears to be growing around the world. Headlines suggest that some places can already meet a large portion of their electricity needs by wind and solar power alone.
The accounting seems straightforward: If wind and solar sources jointly produce more electricity than the amount of electricity consumed, fossil fuel independence seems to be at hand, right?
For obvious reasons, wind and solar energy sources do not provide electricity at a steady rate. If you will, their power delivery is at the whim of nature. Even if their average energy production over a period of time, e.g., a day or a week, equals the average demand over that timeframe, that may not keep the lights on as electricity needs to be produced the very instant it is consumed.
Would installing more solar panels and more wind turbines help provide reliable energy? Could large storage batteries perhaps compensate for the volatility of wind and sun? Could efficiency improvements make a difference in terms of stability?
Future technical innovations are indeed unknowable, but physical laws are firmly established. In general, the efficiency of any system cannot exceed 100%. Specifically for wind turbines, Betz's Law states that the theoretical limit of energy extraction is 59%. With that, firm upper boundaries limit our considerations. Yet, at the same time, they highlight the potential for substantial improvement when compared to the technical realities of today.
With so many unknowns, how can we estimate the long-term potential of renewable energy sources? This is a case for reasoning under uncertainty with Bayesian networks.
As a case study, we explore El Paso, Texas, a city that — at first glance — seems to be a plausible place for exploiting renewable energy sources. It's very sunny and fairly windy, with lots of open space for wind and solar farms.
We also happen to have access to a multi-year dataset of El Paso's hourly electricity usage plus an hourly history of solar radiation and wind speed. Thus, we have the actual conditions in which alternative energy sources would need to perform.
To start the webinar, we demonstrate how to machine-learn probabilistic relationships from historical energy and climate data. Then, we augment this machine-learned Bayesian network with additional nodes that represent the physics of solar panels and wind turbines. Our objective is to explore the boundaries of what might be feasible in the future, given physical and natural constraints.
With such a framework in place, subject matter experts can then provide their beliefs regarding the future efficiency of energy technologies and their costs. Our Bayesian network can take into account these estimated future characteristics of solar panels and wind turbines, plus the anticipated efficiency and cost of traditional power sources or storage batteries, which may still have to supplement the renewables.
Based on learned probabilistic relationships, known physical constants, plus a range of expert assumptions, we can then calculate future scenarios and derive optimal energy mixes for each case. The Bayesian network model of this domain can also help us identify the areas of opportunity by examining the sensitivities of the estimated parameters.
Although this problem domain is undoubtedly important and interesting, we focus on the Bayesian network methodology instead of developing policy recommendations. No part of this webinar should be considered an endorsement or criticism of current renewable energy initiatives.
Stefan Conrady has over 20 years of experience in decision analysis, analytics, market research, and product strategy with Mercedes-Benz, BMW Group, Rolls-Royce Motor Cars, and Nissan, which included assignments in North America, Europe, and Asia.
Today, in his role as Managing Partner of Bayesia USA and Bayesia Singapore, he is recognized as a thought leader in applying Bayesian networks for research, analytics, and reasoning.
Recently, Stefan and his colleague Dr. Lionel Jouffe co-authored Bayesian Networks & BayesiaLab — A Practical Introduction for Researchers, which is now available as an e-book.
The Wisconsin Breast Cancer Database is a well-known dataset for evaluating machine-learned predictive models.
This data was obtained from Dr. William H. Wolberg at the University of Wisconsin Hospitals, Madison.
O. L. Mangasarian and W. H. Wolberg: "Cancer diagnosis via linear programming", SIAM News, Volume 23, Number 5, September 1990, pp 1 & 18.
William H. Wolberg and O.L. Mangasarian: "Multisurface method of pattern separation for medical diagnosis applied to breast cytology", Proceedings of the National Academy of Sciences, U.S.A., Volume 87, December 1990, pp 9193-9196.
O. L. Mangasarian, R. Setiono, and W.H. Wolberg: "Pattern recognition via linear programming: Theory and application to medical diagnosis", in: "Large-scale numerical optimization", Thomas F. Coleman and Yuying Li, editors, SIAM Publications, Philadelphia 1990, pp 22-30.
K. P. Bennett & O. L. Mangasarian: "Robust linear programming discrimination of two linearly inseparable sets", Optimization Methods and Software 1, 1992, 23-34 (Gordon & Breach Science Publishers).
You can find a detailed discussion of this example in of our ebook.
Stefan Conrady, Bayesia USA
Common sense suggests that we always choose the activity with the strongest positive effect on the desired outcome as our top priority for action. For instance, in analyzing call center performance, a statistical model may suggest that the average wait time has the strongest effect on callers' overall satisfaction. With that, and ignoring the cost, for now, we would want to reduce the wait time as our top priority, right? Maybe not.
The critical concept to consider here is joint probability. Unfortunately, in many modeling frameworks, this quantity does not even appear. Hence, any optimization effort would not be able to utilize the joint probability in determining the order of priorities.
Modeling a problem domain with Bayesian networks and BayesiaLab, however, one can calculate the joint probability and use it for optimization purposes. BayesiaLab performs this particular type of optimization with its Target Dynamic Profile function.
In this webinar, we illustrate how we can use Target Dynamic Profile to identify the sequential order in which the key drivers should be improved to maximize overall customer satisfaction.
Stefan Conrady has over 20 years of experience in decision analysis, analytics, market research, and product strategy with Mercedes-Benz, BMW Group, Rolls-Royce Motor Cars, and Nissan, which included assignments in North America, Europe, and Asia.
Today, in his role as Managing Partner of Bayesia USA and Bayesia Singapore, he is recognized as a thought leader in applying Bayesian networks for research, analytics, and reasoning.
Recently, Stefan and his colleague Dr. Lionel Jouffe co-authored Bayesian Networks & BayesiaLab — A Practical Introduction for Researchers, which is now available as an e-book.
Stefan Conrady, Bayesia USA
Given the cost and technical challenges of experimenting with real physical systems, it is common practice in engineering to approximate such domains with computer models. These models incorporate the physical laws, typically in the form of hundreds of differential equations, that govern the system's real-world behavior. That way, the system performance can be examined in detail before building a hardware prototype. Simulations allow engineers to try out different parameters of components in pursuit of performance objectives while taking into account constraints.
While increasing computing power has made simulations more accessible and affordable than ever, performance remains a significant constraint. Why? For each set of parameter values, a new simulation must be run, in which many differential equations need to be solved. While this might only take a few seconds on modern computers, one generally needs to test many levels of many parameters, which would require thousands or even millions of iterations.
So, as the computing requirements grow exponentially with the number of parameters and levels, alternative strategies must be sought. One approach would be to be very selective in terms of which parameters and levels to test, i.e., narrowing the search space by utilizing a search algorithm.
In our webinar, we pursue a different strategy. We approximate the original simulation model, which is based on differential equations, with another model, a so-called meta-model. In effect, we are simulating the simulation of the real-world system. This approach is generally known as Response Surface Methodology, in which the relationships between parameters and target variables are approximated with functions that are unrelated to the underlying physical laws.
In our case, we machine-learn a Bayesian network meta-model from previously computed simulation data and, thus, capture the relationships between parameters and target variables entirely nonparametrically. With such a meta-model, we can perform inference much faster and see the effect of parameter changes in real-time.
In our case study, we examine the impact of tire stiffness, spring rates, and damping constants on the ride comfort of vehicle passengers. We create a so-called "quarter-vehicle model" using a two-mass spring/damper system in Modelica. On that basis, we simulate the vehicle's response to vertical excitations from a synthesized, irregular road surface. This produces the sample observations we need for learning a Bayesian network model with BayesiaLab.
Bayesian networks offer another key advantage in that variables can represent distributions in the frequency domain. This allows for a direct evaluation of the impact of parameter changes on the frequency response of the vertical vehicle body acceleration. This frequency response will be our principal measure for judging ride quality and body control. Generally speaking, we want to minimize peaks in the frequency range between 0 Hz and 200 Hz.
In this context, computing the frequency response curve's entropy, an information-theoretic measure, is very helpful. For instance, BayesiaLab's built-in optimization algorithms can search for parameter values that minimize the entropy of the frequency response, i.e., make the curve as uniform as possible.
Meta models are ultimately shortcuts for a faster system evaluation. As it turns out, we implicitly use meta-models in the qualitative evaluation of systems all the time. Reasoning about a vehicle suspension, we would presumably argue that "softer spring and damper settings" generally provide a "smoother ride," and we could make such a (correct) statement entirely without solving any differential equations.
A recent presentation by Zack Xuereb Conti on simulation meta-modeling with Bayesian networks provided the impetus for developing this case study.
Stefan Conrady has over 20 years of experience in decision analysis, analytics, market research, and product strategy with Mercedes-Benz, BMW Group, Rolls-Royce Motor Cars, and Nissan, which included assignments in North America, Europe, and Asia.
Today, in his role as Managing Partner of Bayesia USA and Bayesia Singapore, he is recognized as a thought leader in applying Bayesian networks for research, analytics, and reasoning.
Recently, Stefan and his colleague Dr. Lionel Jouffe co-authored Bayesian Networks & BayesiaLab — A Practical Introduction for Researchers, which is now available as an e-book.