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In this minisode Kyle and Linh Da discuss the meaning of Bayesian Updating.

The root concept is Bayes Theorem:

\(Pr(A|B) = \dfrac{Pr(B|A) \cdot Pr(A)}{Pr(B)}\)

Although a more helpful way of thinking about this alongside our discussion is to replace \(A\) with \(H\) for Hypothesis, and \(B\) with \(E\) for Evidence.

\(Pr(H|E) = \dfrac{Pr(E|H) \cdot Pr(H)}{Pr(E)}\)

In this case \(Pr(H)\) would represent your probability / belief representing the likelihood that the hypothesis is true. Given new evidence \(E\), you can evaluate how likely that hypothesis was to generate that Evidence (i.e.Â \(Pr(E|H)\)).

In our farmers market example, we hadâ€¦

- \(H_P\) - Box contains 100% (P)omegranates
- \(H_M\) - Box contains (M)ix of 50% pomegranates and 50% lemons
- \(H_L\) - Box contains 100% (L)emons

We established our prior beliefs as \(H_i = \frac{1}{3}\) for \(i=0,1,2\).

When the farm pulls out one example fruit which turns out to be a pomegranate, Linh Da points out that ``itâ€™s not the bad of just lemonsâ€™â€™ (a.k.a. \(Pr(H_L) = 0\)). But that doesnâ€™t mean the other hypothesis are not 50 / 50! What are they?

$ Pr(H_P|E) = = = = 66.% $

$ Pr(H_M|E) = = = = 33.% $

If youâ€™re having a hard time understanding why \(Pr(E) = .50\), itâ€™s because this is your expectation of the evidence you got independent of which hypothesis turns out to be true. Thus, itâ€™s based on your prior (original) belief over each hypothesis and their likelihood of producing the observation. In other wordsâ€¦

$ \[\begin{split} Pr(E) &= Pr(E|H_P) \cdot Pr(H_P) + Pr(E|H_M) \cdot Pr(H_M) + Pr(E|H_L) \cdot Pr(H_L) \\ &= 1.0 \cdot \frac{1}{3} + 0.50 \cdot \frac{1}{3} + 0.0 \cdot \frac{1}{3} \\ &= 0.5 \end{split}\]$