The Teahouse Example

"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."

Will you have a scone with your tea?

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`

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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.

You can experiment with this model in BayesiaLab or via this WebSimulator page: https://simulator.bayesialab.com/#!simulator/160655093718

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.

The Inverse Probability Problem

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)

Will you have tea with your scone?

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%.

Updating Beliefs in Response to Evidence

"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)

To reason about this domain, we first import a small CSV file, which represents Table 3.1 from the book, into BayesiaLab. Tea-Scones.csv

The resulting Bayesian network is available in XBL format here: Teahouse.xbl

"... let $P(T)$denote the probability that a customer orders tea and $P(S)$ denote the probability he orders scones. If we already know a customer has ordered tea, then $P(S | T)$ denotes the probability that he orders scones. (Remember that the vertical line stands for “given that.”) Likewise, $P(T | S)$ denotes the probability that he orders tea, given that we already know he ordered scones ...

$P(S | T) P(T) = P(T | S) P(S)$

The Joint Probability of 41.67% corresponds to the Marginal Likelihood $P(T)$, i.e., the prior probability of a customer ordering a scone.