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  \textbf{\Large{\textsc{STANFORD UNIVERSITY}}}\\[5pt]
  \textbf{\Large{\textsc{DEPARTMENT OF STATISTICS}}}\\[5pt]
  \Large{\textsc{FOUNDATIONS OF STATISTICS SEMINAR}}
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  3:15 p.m., Wednesday, November 21, 2007\\
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  Building 380, Room 380-C\\
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  \textsl{Terrence L. Fine} \\
  School of Electrical \& Computer Engineering\\
  Cornell University\\
  Ithaca, NY 14850
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  \subsection*{Chaotic Probability: The Set of Measures Model}
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\noindent

We are motivated by the success of a sets of measures approach to representing the state of belief
of an individual, as expressed through upper and lower previsions/expectations, to ask whether such
a model can also describe some objective frequentist data sources.

I report on joint work with Pablo I. Fierens and Leandro C. Rego on a probability model for
sequences in which the conditional probability $\nu_i$ of the $i$-th random variable $X_i$,
conditional upon all preceding random variables $X^{i-1} = x^{i-1}$, is chosen erratically from a
set of measures $\mathcal{M}= \{\nu\}$, all being defined on the same finite sample space
$\mathcal{X}$. Our interest is in the case where the function $F(x^{i-1})$ selecting the marginal
$\nu_i$ is not effectively computable and not asymptotically estimable from the observed data
$x^i$. We can think of this setup as arising from a game between two agents, although we are more
interested in the possibility that the physical world provides examples of such erratic or highly
irregular processes.

Our goal is to estimate $\mathcal{M}$ by $\hat{\mathcal{M}}$ based upon $x^n$, for large enough
$n$. We show the existence of a family $\Psi= \{\varphi\}$ of causal place selection rules, chosen
independently of $\mathcal{M}$, that produce subsequences of the given data whose time averages
provide the desired information about the elements of $\mathcal{M}$. We have not been able to show
the converse that $\mathcal{M}$ is all that can be estimated.

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