# A categorical foundation for Bayesian probability

- Published in 2012
- Added on

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Given two measurable spaces $H$ and $D$ with countably generated $\sigma$-algebras, a prior probability measure $P_H$ on $H$ and a sampling distribution $\mcS:H \rightarrow D$, there is a corresponding inference map $\mcI:D \rightarrow H$ which is unique up to a set of measure zero. Thus, given a data measurement $\mu:1 \rightarrow D$, a posterior probability $\hat{P_H}=\mcI \circ \mu$ can be computed. This procedure is iterative: with each updated probability $P_H$, we obtain a new joint distribution which in turn yields a new inference map $\mcI$ and the process repeats with each additional measurement. The main result shows that the assumption of Polish spaces to obtain regular conditional probabilities is not necessary---countably generated spaces suffice. This less stringent condition then allows for non-trivial decision rules (Eilenberg--Moore algebras) on finite (as well as non finite) spaces, and also provides for a common framework for decision theory and Bayesian probability.

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## Other information

- arxivId
- 1205.1488
- keywords
- bayesian probability,categorical foundation for probability,giry,monad,msc 2000 subject,primary 60a05,probabilistic logic,regular conditional probability,secondary 62c10
- pages
- 18

### BibTeX entry

@article{Culbertson2012, title = {A categorical foundation for Bayesian probability}, author = {Culbertson, Jared and Sturtz, Kirk}, url = {http://arxiv.org/abs/1205.1488 http://arxiv.org/pdf/1205.1488v3}, abstract = {Given two measurable spaces {\$}H{\$} and {\$}D{\$} with countably generated {\$}\sigma{\$}-algebras, a prior probability measure {\$}P{\_}H{\$} on {\$}H{\$} and a sampling distribution {\$}\mcS:H \rightarrow D{\$}, there is a corresponding inference map {\$}\mcI:D \rightarrow H{\$} which is unique up to a set of measure zero. Thus, given a data measurement {\$}\mu:1 \rightarrow D{\$}, a posterior probability {\$}\hat{\{}P{\_}H{\}}=\mcI \circ \mu{\$} can be computed. This procedure is iterative: with each updated probability {\$}P{\_}H{\$}, we obtain a new joint distribution which in turn yields a new inference map {\$}\mcI{\$} and the process repeats with each additional measurement. The main result shows that the assumption of Polish spaces to obtain regular conditional probabilities is not necessary---countably generated spaces suffice. This less stringent condition then allows for non-trivial decision rules (Eilenberg--Moore algebras) on finite (as well as non finite) spaces, and also provides for a common framework for decision theory and Bayesian probability.}, archivePrefix = {arXiv}, arxivId = {1205.1488}, eprint = {1205.1488}, keywords = {bayesian probability,categorical foundation for probability,giry,monad,msc 2000 subject,primary 60a05,probabilistic logic,regular conditional probability,secondary 62c10}, month = {may}, pages = 18, year = 2012, urldate = {2012-05-08}, collections = {Probability and statistics} }