Multidimensional%20Integration%20Part%20I

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Multidimensional IntegrationPart I

• Harrison B. Prosper
• Florida State University

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Outline

• Do we need it?
• Markov Chain Monte Carlo
• Adaptive Methods
• Summary

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Do we need it?

• Most analyses in high energy physics are done
  using frequentist methods.
• The more sophisticated ones typically involve the
  minimization of log likelihoods using programs
  such as the celebrated MINUIT.
• These methods in general do not need
  multidimensional integration.

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But we may need it if

• We wish to do analyses using Bayesian methods.
  Here are a few examples
• Limit-setting
• DØ Single Top Analysis
• Luminosity estimation
• SUSY/Higgs Workshop
• Jet energy scale corrections

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Jet Energy Scale Corrections
1. Assume we have a pure sample of
2. Assume
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A single event
Likelihood
Prior
Posterior
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But need lots of events, in practice!
Posterior
Number of dimensions 2Nm, where N is the
number of events used and m is the number of a
parameters. If N 1000 and m 3, we have Ndim
2000!!
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Multidimensional Integration

• Low dimensions, that is, lt than about 20
• Adaptive Numerical Integration
• Recursively partition space while working to
  reduce the integration error on the partition,
  which at a given step, has the largest error.
• High dimensions, that is, gt than about 20
• Markov Chain Monte Carlo

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The Basic Idea

• Generate a sequence of parameter values ai from
  the posterior distribution Post(aD) and compute
  averages
• In general, it its very difficult to sample
  directly from a complicated distribution.
  Gaussians, of course are easy!
• Must use indirect method to generate sequence.

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The Basic Idea, cont.

• If the sequence of ai are statistically
  independent the uncertainty in the estimate of
  the integral is just the error on the mean
• Important The error reduces slowly, but it does
  so in a manner that is independent of the
  dimensionality of the space.

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Markov Chain Monte Carlo

• State, x a vector of real-valued quantities
• Transition probability, T probability to get
  state x(t1) given state x(t)
• Proposal probability, q probability to propose a
  new state y(t1) given state x(t)
• Acceptance probability, A probability to accept
  the proposed state.
• Markov chain random sequence of states x(t) with
  the property that the probability to get state
  x(t1) depends only on the previous state x(t).

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MCMC 1

• Let pt1(x) be the probability of state x at time
  step t1 and pt(x) be the probability of state x
  at time step t. Then
• The goal is to produce the following condition
• as the time step t goes to infinity. That is, to
  arrive at a stationary (or invariant, or
  equilibrium) distribution p(x)

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MCMC 2

• Next time we shall see how that condition can be
  achieved!