Bayesian Revision of Probabilities


Bayesian Revision with Probability Trees
Probability trees make revising probability distributions a snap. Two trees are required. The first tree displays the input data (the prior probability distribution and the likelihood function) along with the joint distribution of the random variables, obtained by simple multiplication:

Probability Tree


Note that probability trees are not decision trees (just as palm trees are not pine trees). The nodes of a probability tree all refer to probability distributions, which means that neither decision nodes (squares) nor states of nature nodes (circles) are used.


The second tree, called the flip tree, is constructed as follows:
 
1. Place the joints in their new positions at the end of the tree's branches (be careful, the ordering changes).*
2. Compute the preposteriors by summing the joints emanating from that preposterior's branch.
3. Compute the posteriors by dividing each joint by its corresponding preposterior.
 
* Example:  P( S ) · P( F | S )  =  P( S ∩ F )  =  P( F ) · P( S | F )

Flip Tree


Example:  P( H | FH )  =  P( H ∩ FH ) / P( FH )  =  0.07 / 0.19  ≈  0.3684
 
The flip tree visually shows why the preposteriors have that name: they come just before the posteriors.

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