Abstract
AbstractA desired closure property in Bayesian probability is that an updated posterior distribution be in the same class of distributions – say Gaussians – as the prior distribution. When the updating takes place via a statistical model, one calls the class of prior distributions the ‘conjugate priors’ of the model. This paper gives (1) an abstract formulation of this notion of conjugate prior, using channels, in a graphical language, (2) a simple abstract proof that such conjugate priors yield Bayesian inversions and (3) an extension to multiple updates. The theory is illustrated with several standard examples.
Highlights
The main result of this paper, Theorem 6.3, is mathematically trivial
As described in the introduction, the informal definition says that a class of distributions is conjugate prior to a statistical model if the associated posteriors are in the same class of distributions
This paper contains a novel view on conjugate priors, using the concept of channel in a systematic manner
Summary
The main result of this paper, Theorem 6.3, is mathematically trivial. But it is not entirely trivial to see that this result is trivial. The effort and contribution of this paper lie in setting up a framework – using the abstract language of channels, Kleisli maps and string diagrams for probability theory – to define the notion of conjugate prior in such a way that there is a trivial proof of the main statement, saying that conjugate priors yield Bayesian inversions. It comes close to our channel-based description, since it explicitly mentions the conjugate family as a conditional probability distribution with (recomputed) parameters. What has been left unexplained is the ‘suitable’ equation that the parameter translation function h : P × O → P should satisfy It is not entirely trivial, because it is an equation between channels in what is called the Kleisli category K (G) of the Giry monad G.
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