A categorical foundation for Bayesian probability
- Published in 2012
- Added on
In the collection
Given two measurable spaces $H$ and $D$ with countably generated $\sigma$-algebras, a prior probability measure $P_H$ on $H$ and a sampling distribution $\mathcal{S}:H \rightarrow D$, there is a corresponding inference map $\mathcal{I}:D \rightarrow H$ which is unique up to a set of measure zero. Thus, given a data measurement $\mu : 1 \rightarrow D$, a posterior probability $\hat{P_H}=\mathcal{I} \circ \mu$ 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 $\mathcal{I}$ 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
- key
- Culbertson2012
- type
- article
- date_added
- 2012-05-08
- date_published
- 2012-05-01
- 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, key = {Culbertson2012}, type = {article}, title = {A categorical foundation for Bayesian probability}, author = {Culbertson, Jared and Sturtz, Kirk}, 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 {\$}\mathcal{\{}S{\}}:H \rightarrow D{\$}, there is a corresponding inference map {\$}\mathcal{\{}I{\}}: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{\}}=\mathcal{\{}I{\}} \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 {\$}\mathcal{\{}I{\}}{\$} 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.}, comment = {}, date_added = {2012-05-08}, date_published = {2012-05-01}, urls = {http://arxiv.org/abs/1205.1488,http://arxiv.org/pdf/1205.1488v3}, collections = {probability-and-statistics}, url = {http://arxiv.org/abs/1205.1488 http://arxiv.org/pdf/1205.1488v3}, urldate = {2012-05-08}, year = 2012, 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 }