022807 - Lecture 11 Recall Getting to Metropolis algorithm...

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Lecture 11: 02/28/2007 Recall: Getting to Metropolis algorithm Problem: X= phase space for a physical system of “many particles” (huge) Have a probability distribution on states: Boltzmann or Gibbs distribution: = -∈( ) px C0e x kT T=temp K=Boltzmann’s constant (x)=??? Of states Є Want to “draw a sample according to p(x)”—ie. Generate a population of points in X distributed roughly according to p(x). (Rationale : good for computing E p (f(x)) where f(x) is a “physical variable of interest” by SLLN) P(x) might be like (for some given T). Figure1 Intuitively want population to be concentrated under peaks in p(x). To understand algorithm better, imagine we’re working with a fixed discretization of X—call the set of points in discretization S. Metropolis Algorithm : pick some point in S j call it Θ D . Go to step (1); store Θ 0 in P(a big multi-set of points in X) 1. Assume you have Θ m . From Ψ m by perturbing Θ m according to a uniform * random variable.
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  • Spring '07
  • Algorithms, Probability theory, Stochastic process, Markov chain, Quantification, Universal quantification, Markov chain Monte Carlo

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