I was hoping I could produce a one-to-one mapping between diagrams of overlapping factors (from FFS) and Bayes nets. This is not possible, but instead I have been able to produce a similar mapping between factor overlap diagrams and sets of Bayes nets, subject to certain compatibility relationships.

Bayes Nets

The Bayes nets over some variables <span>_X_</span> can be thought of as a category of objects related by "bookkeeping" (and similar) relationships. To be more specific, we can say that a unique morphism exists between two Bayes nets <span>_N_i rightarrow N_j_</span> exists if and only if, for all probability distributions <span>_P_</span> over the elements of <span>_N_i_</span>, the approximation of <span>_N_j_</span> over <span>_P_</span> is no worse than the approximation of <span>_N_i_</span> over <span>_P_</span>. For two variables we need the Markov-chain-re-rooting and arrow-adding rules:

We could (with some effort) do three variables, but since there are three options for each pair [...]

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Outline:

(00:22) Bayes Nets

(01:48) From Bayes Nets to Equivalence Classes

(04:32) Finite Factored Sets

(06:04) Mapping Them Together

(07:31) From Equivalence Classes to Compatible Sets

(10:26) A few Illustrative Examples

(11:11) Conjectures and Limitations

(13:25) Proofs

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First published:
January 23rd, 2024

Source:
https://www.lesswrong.com/posts/atBjEzXaD6xSWmFhy/from-finite-factors-to-bayes-nets

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Narrated by TYPE III AUDIO.

“From Finite Factors to Bayes Nets” by J Bostock

LessWrong (30+ Karma)