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Multidimensional Integration Part I

Explore the need for multidimensional integration in high-energy physics analyses. Learn about Markov Chain Monte Carlo and Adaptive Methods for Bayesian analysis, including examples like Limit-setting and Jet Energy Scale Corrections. Discover the basics of generating parameter sequences from posterior distributions and the concept of Markov Chain Monte Carlo for complex distribution sampling.

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Multidimensional Integration Part I

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  1. Multidimensional IntegrationPart I Harrison B. Prosper Florida State University Fermilab RAAG Meeting, 25 May, 2001, Harrison B. Prosper

  2. Outline • Do we need it? • Markov Chain Monte Carlo • Adaptive Methods • Summary Fermilab RAAG Meeting, 25 May, 2001, Harrison B. Prosper

  3. 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. Fermilab RAAG Meeting, 25 May, 2001, Harrison B. Prosper

  4. 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 Fermilab RAAG Meeting, 25 May, 2001, Harrison B. Prosper

  5. Jet Energy Scale Corrections 1. Assume we have a pure sample of 2. Assume Fermilab RAAG Meeting, 25 May, 2001, Harrison B. Prosper

  6. A single event… Likelihood Prior Posterior Fermilab RAAG Meeting, 25 May, 2001, Harrison B. Prosper

  7. But need lots of events, in practice! Posterior Number of dimensions = 2N+m, 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!! Fermilab RAAG Meeting, 25 May, 2001, Harrison B. Prosper

  8. Multidimensional Integration • Low dimensions, that is, < 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, > than about 20 • Markov Chain Monte Carlo Fermilab RAAG Meeting, 25 May, 2001, Harrison B. Prosper

  9. The Basic Idea • Generate a sequence of parameter values ai from the posterior distribution Post(a|D) 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. Fermilab RAAG Meeting, 25 May, 2001, Harrison B. Prosper

  10. 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. Fermilab RAAG Meeting, 25 May, 2001, Harrison B. Prosper

  11. Markov Chain Monte Carlo • State, x: a vector of real-valued quantities • Transition probability, T: probability to get state x(t+1) given state x(t) • Proposal probability, q: probability to propose a new state y(t+1) 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(t+1) depends only on the previous state x(t). Fermilab RAAG Meeting, 25 May, 2001, Harrison B. Prosper

  12. MCMC 1 • Let pt+1(x) be the probability of state x at time step t+1 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 π(x) Fermilab RAAG Meeting, 25 May, 2001, Harrison B. Prosper

  13. MCMC 2 • Next time we shall see how that condition can be achieved…! Fermilab RAAG Meeting, 25 May, 2001, Harrison B. Prosper

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