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\usepackage{amsmath}
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\usepackage{verbatim}
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%
\newcommand{\field}[1]{\mathbb{#1}}
\DeclareMathOperator{\PR}{\field{P}}             % Probability
\DeclareMathOperator{\E}{\field{E}}              % Expectation
\def\N{\field{N}}                                % nonnegative integers
\def\R{\field{R}}                                % Blackboard R - Reals
\def\F{\field{F}}                                % Functions/stationary policies
\def\A{\field{A}}                                % Action space
\def\X{\field{X}}                                % State space

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\begin{document}
  \Talktitle{
%%
A Bayesian Method to uncertain MDPs by Markov set-chains
%%
}
  \Presentingauthor{Masayuki Horiguchi}, 
\Affiliation{Kanagawa University, Japan}, 
\email{horiguchi@kanagawa-u.ac.jp}
  \Coauthor{Masami Yasuda},
\Affiliation{Chiba University, Japan},
\email{yasuda@math.s.chiba-u.ac.jp}
  \Coauthor{Masami Kurano}, 
\Affiliation{Chiba University, emeritus, Japan},
\email{kurano@seaple.ne.jp}
\abstract{
% (Not more than 250 words.)
We are concerned with Bayesian inference of transition law in MDPs.
Applying De Robertis/Hartigan's result(1981, Ann.~Math) of Bayesian 
inference on
intervals of measure for MDPs, it is formulated as uncertain MDPs 
with intervals of prior measures.
We characterize the uncertain model as a controlled 
Markov set-chain model with intervals of transition matrices.
Here we adapted the notion of Markov se-chain 
by Hartfiel(1998, Lec.~Note in Math). 

 Firstly a model construction to MDP is given by a set of interval 
matrices. The true transition matrix is estimated by an interval matrix
and then uncertain MDPs of Bayesian approach is defined by a controlled 
Markov set-chain model. The sufficient and necessary condition that
a policy is Pareto-optimal by an optimality inclusion relation.
By using Bayesian inference on intervals of measure for uncertain MDPs,
interval matrices of posterior measures are given.
Some convergence property for uncertain MDPs of Bayesian approach
is discussed and also a numerical experiment is shown.
}
\end{document}

\section{Formulation and Notations}

\section{Bayesian inference for interval of prior measures}

\section{Continuity and convergence of uncertain MDPS}

\section{Numerical experiment}


\begin{thebibliography}{99}
\bibitem{yas} aa.

\bibitem{bell}R.E.~Bellman, {\em Dynamic Programming}, Princeton Univ. Press, NJ,  1957.


\end{thebibliography}

\end{document}

