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Jorge Soto-Andrade (Dec 26 2023 at 15:35):Hi! I have a naive question on a categorical approach to Bayes. Say, the classical Bayesian setting where you have a bunch of urns with coloured balls, with different compositions, you choose blindly an urn at random, draw a ball, get Black and wonder about the probability of having chosen such and such urn.
I wonder whether the following remark is well known:
what you have in the Bayesian situation
Bayes_urnes_full_flow_ETM6.jpg.pdf
is a stochastic mapping from the set U of urns to the set C of colours. Bayes calculates the "dual" stochastic mapping from C to U. You have then a sort of "reversible" stochastic process: if you feed it with a random urn, you get a random colour, if you apply the dual stoch mapping to this random colour you get back your random urn. Bayesian updating boils down to figuring out which random urn you get if you feed the Bayesian machine with a random colour like "Black for sure". I enclose an iconic rendering of this in a toy example (two urns, two colours), corresponding
Bayes_urnes_pedestrian_etm6.pdf
to a fluid approach (hydraulic metaphor activated) to Bayes, and also a "pedestrian approach".
Does this ring a categorical bell to anyone?
Jorge Soto-Andrade (Dec 26 2023 at 15:41):PS In the fluid iconic rendering the stoch mapping going from urns to colours is given by the four blue diagonal weighted arrows, the dual stochastic mapping is given by the four green diagonal weighted arrows. The horizontal and vertical weighted arrows represent the initial random urn and the ensuing random colour. This could be an example of diagrammatic linear algebra too...
John Baez (Dec 27 2023 at 10:32):You seem to be sketching the usual categorical approach to Bayes' theorem! You might like this:
Tobias Fritz (Dec 27 2023 at 11:08):Actually I wouldn't recommend that paper. It's been one of the earlier ones on the topic, but it's confusingly written, and their formalism has lots of extra structure that is not needed to talk about Bayesian updating (or anything else in probability), and that extra structure does not exist once you go beyond discrete probability with finite sets. An improved account of Bayesian updating is given here:
Jorge Soto-Andrade (Dec 27 2023 at 14:50)::pray: :pray:
Elena Di Lavore (Jan 12 2024 at 10:38):you might be also interested in theorem 3.28 in Evidential decision theory via partial Markov categories
