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One of Grothendieck's dicta about algebraic geometry is to consider "the relative situation", where one doesn't consider the category of schemes but of schemes over a fixed base scheme.

In Bayesian probability, one computes probabilities of events given new knowledge, but only upon having specified a "prior".

Is there any categorical interpretation of the latter process, where specifying the prior is visibly analogous to specifying a base scheme?

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    $\begingroup$ I dunno, man. Here's a toy model of Bayesian updating: your state of knowledge about the world is represented by a subset $S$ of the set $X$ of states the world could be in. Someone tells you a fact about the world; now you've learned that the state of the world lies in a subset $T$. Your new updated state of knowledge is $S \cap T$. Is there an interesting category structure here? Maybe inclusion of subsets, but that's not so interesting. $\endgroup$
    – Qiaochu Yuan
    Commented Nov 17, 2015 at 2:24
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    $\begingroup$ I remember a talk by Alex Simpson on categorical treatment of conditional independence, where some sort of things-over-a-base were essential. He referred to a question here on MO; one of the answers to this question opens with a link to the paper entitled "A Categorical Foundation for Bayesian Probability". I have not seen it but maybe... $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Nov 17, 2015 at 21:11
  • $\begingroup$ @QiaochuYuan Should your comment say that a boring toy model of Bayesian updating is related to a boring toy model of categories? But wait, inclusion of sets in not really a toy model for categories, so… $\endgroup$
    – Dirk
    Commented Nov 18, 2015 at 8:49

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In statistics in general one considers a class of 'model(s)'. Usually this model has a number of parameters. For example if the model was 'event distribution is gaussian' then the parameters would be the mean and standard deviation. So there is a space of models. Usually the game is to find the model that best fits the observed data (find the mle). In Bayesian statistics, there is the added tweak of a prior distribution of the probabilities of obtaining (observing?) the various parameters. How does on obtain the priors ? That is, to the extent I understand it, the 64k question. However, I could be convinced that any given measure on the space of all priors would constituted a base scheme. I find that Efron's article (published in BAMS !) has as an excellent explanation of Bayesian thought. I've posted this as an answer due to it's length, though perhaps it is an answer !

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