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

2. 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

3. 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

4. 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

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

10. 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

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. 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

13. 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