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\begin{document}

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\section*{monads of probability, measures, and valuations}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{measure_and_probability_theory}{}\paragraph*{{Measure and probability theory}}\label{measure_and_probability_theory}

[[!include measure theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{functor_unit_and_multiplication}{Functor, unit and multiplication}\dotfill \pageref*{functor_unit_and_multiplication} \linebreak
\noindent\hyperlink{algebras}{Algebras}\dotfill \pageref*{algebras} \linebreak
\noindent\hyperlink{kleisli_morphisms}{Kleisli morphisms}\dotfill \pageref*{kleisli_morphisms} \linebreak
\noindent\hyperlink{monoidal_structure}{Monoidal structure}\dotfill \pageref*{monoidal_structure} \linebreak
\noindent\hyperlink{duality}{Duality}\dotfill \pageref*{duality} \linebreak
\noindent\hyperlink{detailed_list}{Detailed list}\dotfill \pageref*{detailed_list} \linebreak
\noindent\hyperlink{see_also}{See also}\dotfill \pageref*{see_also} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In many categorical approaches to [[measure theory]] and [[probability]], one considers a [[category]] of [[spaces]], such as [[measurable spaces]] or [[topological spaces]], and equips this category with a [[monad]] whose [[functor]] part assigns to each space $X$ a space $P X$ of [[measures]], [[probability measures]], or [[valuation (measure theory)|valuations]] on $X$, or a variation thereof.

For probability theory, this can be interpreted as adding to the [[points]] of a space $X$ new ``random points'', modelled as probability measures or valuations. The old points, which we can think of as deterministic, are embedded in $P X$ via the unit of the monad $X\to P X$. Just as well, the [[Kleisli morphisms]] of $P$ can be seen as [[stochastic maps]]. (Monads can be seen as ways of extending our spaces and functions to account for new phenomena, see for example [[extension system]] and [[monad in computer science]].) Note that these probability measures are technically different from [[random elements]]: they rather correspond to the [[law of a random variable|laws]] of the random elements.

[[algebra over a monad|Algebras]] of probability and measure monads can be interpreted as generalized [[convex spaces]] or [[conical spaces]] of a certain kind. For probability theory, in particular, the algebras of a probability monad can be seen as spaces equipped with a notion of [[expectation value]] of a [[random variable]]. The details vary depending on the monad and on the category under consideration.

Many choices of categories and of monads are possible, depending on which aspects of measure theory or probability one wants to study. See \hyperlink{detailed_list}{the table below} for more details.

The term ``probability monad'' was coined by Giry herself (see \hyperlink{giry80}{here}).

\hypertarget{functor_unit_and_multiplication}{}\subsection*{{Functor, unit and multiplication}}\label{functor_unit_and_multiplication}

(\ldots{}work in progress\ldots{})

\hypertarget{algebras}{}\subsection*{{Algebras}}\label{algebras}

(\ldots{}work in progress\ldots{})

\hypertarget{kleisli_morphisms}{}\subsection*{{Kleisli morphisms}}\label{kleisli_morphisms}

(\ldots{}work in progress\ldots{})

\hypertarget{monoidal_structure}{}\subsection*{{Monoidal structure}}\label{monoidal_structure}

(\ldots{}work in progress\ldots{})

\hypertarget{duality}{}\subsection*{{Duality}}\label{duality}

(\ldots{}work in progress\ldots{})

\hypertarget{detailed_list}{}\subsection*{{Detailed list}}\label{detailed_list}

(\ldots{}to be expanded\ldots{})

\hypertarget{see_also}{}\subsection*{{See also}}\label{see_also}

\begin{itemize}%
\item [[monad]], [[algebra over a monad]]
\item [[monad (in computer science)]]
\item [[Giry monad]]
\item [[probability theory]]
\item [[measure theory]]
\item [[valuation (measure theory)]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[W. Lawvere]], \emph{The category of probabilistic mappings}, ms. 12 pages, 1962 ([[lawvereprobability1962.pdf:file]])


\item Mich\`e{}le Giry, \emph{A categorical approach to probability theory}, Categorical aspects of topology and analysis (Ottawa, Ont., 1980), pp. 68--85, Lecture Notes in Math. \textbf{915} Springer 1982.


\item T. Swirszcz, \emph{Monadic functors and convexity}, Bulletin de l'Academie Polonais des Sciences 22, 1974 (\href{https://www.fuw.edu.pl/~kostecki/scans/swirszcz1974.pdf}{pdf})


\item [[Klaus Keimel]], \emph{The monad of probability measures over compact ordered spaces and its Eilenberg-Moore algebras}, Topology and its Applications, 2008 (\href{https://doi.org/10.1016/j.topol.2008.07.002}{doi:10.1016/j.topol.2008.07.002})


\item Reinhold Heckmann, \emph{Spaces of valuations}, Papers on General Topology and Ap-plications, 1996 (\href{https://doi.org/10.1111/j.1749-6632.1996.tb49168.x}{doi:10.1111/j.1749-6632.1996.tb49168.x},\href{http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.45.5845&rep=rep1&type=pdf}{pdf})


\item Mauricio Alvarez-Manilla, Achin Jung, [[Klaus Keimel]], \emph{The probabilistic powerdomain for stably compact spaces}, Theoretical Computer Science 328, 2004. \href{https://www.sciencedirect.com/science/article/pii/S0304397504004074}{Link here}.


\item C. Jones and [[Gordon. D. Plotkin]], \emph{A probabilistic powerdomain of evaluations}, LICS 4, 1989. (\href{https://doi.org/10.1109/LICS.1989.39173}{doi:10.1109/LICS.1989.39173})


\item [[Jean Goubault-Larrecq]] and Xiaodong Jia, \emph{Algebras of the extended probabilistic powerdomain monad}, ENTCS 345, 2019 (\href{https://doi.org/10.1016/j.entcs.2019.07.015}{doi:10.1016/j.entcs.2019.07.015})


\item [[Tobias Fritz]], Paolo Perrone and Sharwin Rezagholi, \emph{Probability, valuations, hyperspace: Three monads on Top and the support as a morphism}, 2019. \href{https://arxiv.org/abs/1910.03752}{Link here}.


\item [[Steve Vickers]], \emph{A monad of valuation locales}, 2011. \href{https://www.cs.bham.ac.uk/~sjv/Riesz.pdf}{Link here}.


\item Franck van Breugel, \emph{The metric monad for probabilistic nondeterminism}, unpublished, 2005. (\href{http://www.cse.yorku.ca/~franck/research/drafts/monad.pdf}{pdf})


\item [[Tobias Fritz]] and Paolo Perrone, \emph{A probability monad as the colimit of spaces of finite samples}, Theory and Applications of Categories 34, 2019. (\href{http://www.tac.mta.ca/tac/volumes/34/7/34-07.pdf}{pdf})


\item [[Tobias Fritz]] and Paolo Perrone, \emph{Stochastic order on metric spaces and the ordered Kantorovich monad}, submitted, 2018. (\href{https://arxiv.org/abs/1808.09898}{arXiv:1808.09898})


\item [[Bart Jacobs]], \emph{From probability monads to commutative effectuses}, Journal of Logical and Algebraic Methods in Programming 94, 2018. (\href{https://doi.org/10.1016/j.jlamp.2016.11.006}{doi:10.1016/j.jlamp.2016.11.006})


\item [[Tobias Fritz]], \emph{Convex spaces I: definitions and examples}, 2009. (\href{https://arxiv.org/abs/0903.5522}{arXiv:0903.5522})



\end{itemize}
[[!redirects probability monad]] [[!redirects probability monads]] [[!redirects measure monad]] [[!redirects measure monads]] [[!redirects valuation monad]] [[!redirects valuation monads]] [[!redirects monad of probability]] [[!redirects monads of probability]] [[!redirects monad of probability measures]] [[!redirects monads of probability measures]] [[!redirects monad of measures]] [[!redirects monads of measures]] [[!redirects monad of valuations]] [[!redirects monads of valuations]] [[!redirects probability and measure monad]] [[!redirects probability and measure monads]] [[!redirects measure and valuation monad]] [[!redirects measure and valuation monads]] [[!redirects monads of probability measures valuations]] [[!redirects monads of probability measures and valuations]] [[!redirects monads of probability, measures and valuations]] [[!redirects monads of probability, measures, valuations]]



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