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

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

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

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

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

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

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

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

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

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

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

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

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