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Prediction Processes; Markovian (and Conceivably Causal) Representations of Stochastic Processes

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This descends from the slides for a talk I've been giving, in some form, since working on the papers that became my dissertation in the late 1990s. We'll try to construct such states by constructing predictions. (Discrete time is not strictly necessary, but cuts down on measure theory.) When we make such a guess, we are going to attend to selected aspects of 𝑋 − ∞ 𝑡 (mean, variance, phase of 1st three Fourier modes, ...). This means that our guess is a function, or formally a statistic, of 𝑋 − ∞ 𝑡 . What's a good statistic to use? At this point, the only sane response is "so what?", perhaps followed by making up bizarre-looking functionals and defining statistics to be shmufficient if they maximize those functionals. There is however a good reason to care about sufficiency, embodied in a theorem of Blackwell and Girshick: under any loss function, the optimal strategy can be implemented using only knowledge of a sufficient statistic --- the full data are not needed. For reasonabl

Prediction Processes; Markovian (and Conceivably Causal) Representations of Stochastic Processes Notebooks Prediction Processes; Markovian (and Conceivably Causal) Representations of Stochastic Processes Last update : 07 Jul 2025 12:14 First version : 1 August 2016 \[ \newcommand{\indep}{\mathrel{\perp\llap{\perp}}} \newcommand{\Prob}[1]{\mathrm{Pr}\left( #1 \right)} \] This descends from the slides for a talk I've been giving, in some form, since working on the papers that became my dissertation in the late 1990s. 0. "State" In classical physics or dynamics, the state of a system is the prese

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