The Teahouse Example

"To see how Bayesโ€™s method works, letโ€™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."

Pearl, Judea. The Book of Why: The New Science of Cause and Effect (pp. 99-100). Basic Books. Kindle Edition.

Will you have a scone with your tea?

  • 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

  • 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.

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

  • 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

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

P(SโˆฃT)P(T)=P(TโˆฃS)P(S)P(S | T) P(T) = P(T | S) P(S)

This innocent-looking equation came to be known as โ€œBayesโ€™s rule.โ€ 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%.

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

Updating Beliefs in Response to Evidence

"We can also look at Bayesโ€™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, Bayesโ€™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 Bayesโ€™s rule not just as a convenient definition of the new concept of โ€œconditional probabilityโ€ but as an empirical claim to faithfully represent the English expression โ€œgiven that I know.โ€ (Pearl, pp. 101-102)

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