Paradoxes and Fallacies
There are a number of paradoxes and fallacies that keep recurring as popular and mind-bending puzzles in the media. Although there is (now) complete agreement among scientists on how to resolve them, the correct answers are often perplexing to the casual observer and still cause bewilderment.
We will start off with the fallacy of the transposed conditional, which has become rather infamous and is better known as Prosecutor’s Fallacy. As the name implies, it is a problem often encountered in courts of law, and there are numerous cases of incorrect convictions as a result of this fallacy.
No less serious are the potential consequences of Simpson’s Paradox, for instance, when determining the treatment effect of a new drug under study. The effect of a drug on two subgroups may appear as the complete opposite of the treatment effect on the whole group.
Monty Hall Problem
On a much lighter note, the Monty Hall Problem has its origin in a television game show and might perhaps be the most difficult puzzle to comprehend intuitively, even when explicit proof is provided. Respected mathematicians and statisticians have struggled with this problem, and some of them have boldly proclaimed wrong solutions.
The counterintuitive nature of these probabilistic problems relates to the cognitive limits of human inference. More specifically, we are dealing with the problem of updating beliefs given new evidence, i.e. carrying out inference. This cognitive challenge may seem surprising, given that humans are exceptionally gifted in discovering causal structures in their everyday environment. Discovering causality in the world is quite literally child’s play, as babies start understanding the world through a combination of observation and experimentation. Our human intuition is actually quite good when it comes to reasoning from cause to effect; our qualitative perception of such relationships (even under uncertainty) is often compatible with formal computations.
Reasoning Under Uncertainty
However, when it comes to reasoning under uncertainty in the opposite direction, from effect to cause, i.e. diagnosis, or when combining multiple pieces of evidence, conventional wisdom frequently fails catastrophically. Even worse, the correct inference in such situations is often completely counterintuitive to people and feels utterly wrong to them. It is not an exaggeration to say that their sense of reason is violated.
For more traditional computations, such as arithmetics, we have many tools that help us address our mental shortcomings. For instance, we can use paper and pencil to add 9,263,891 and 1,421,602 as most of us can’t do this in our heads. Alternatively, we can use a spreadsheet for this computation. In any case, it will not surprise us that the sum of those two numbers is a little over 10.5 million. The computed result is entirely consistent with our intuition.
As this paper will show, the formally correct solutions of these probabilistic paradoxes are counterintuitive. In addition to being counterintuitive, there are few tools assisting us in solving them. There is no spreadsheet that allows us to simply plug in the numbers to calculate the result.
Bayes' Rule & Bayesian Networks
Although we won’t be able to overcome inherent mental biases and cognitive limitations, we can now provide a very practical new tool for the correct inference in the form of Bayesian networks. Bayesian networks derive their name from Reverend Thomas Bayes, who, in the middle of the 18th century, first stated the rule for computing inverse probabilities.
Bayesian networks offer a framework that allows applying Bayes’ Rule for updating beliefs in the same way spreadsheets are very convenient for applying arithmetic operations to many numbers. We will show how restating these vexing problem domains as simple Bayesian networks offers near-instant solutions. Just as spreadsheets help us perform arithmetic operations externally, i.e. outside our head, Bayesian networks offer a reliable structure to precisely perform inferential computations, which we can’t manage in our minds. The visual nature of Bayesian networks furthermore helps (at least a little) in making these paradoxes more intuitive to our own human way of thinking.
Beyond utilizing Bayesian networks as the framework, we will use BayesiaLab as the software tool for network creation, editing and inference. This allows us to leverage all the theoretical benefits of Bayesian networks for practical use via an intuitive graphical user interface.