Bayesian Neural Networks Pushpa Bhat Fermilab Harrison Prosper Florida State University

9/14/05PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper Outline  Introduction  Bayesian Learning  Simple Examples  Summary

9/14/05PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper Multivariate Methods  Since the early 1990’s, we have used multivariate methods extensively in Particle Physics  Some examples  Particle ID and signal/background discrimination  Optimization of cuts for top quark discovery at DØ  Precision measurement of top mass  Searches for leptoquarks, technicolor,..  Neural network methods have become popular due to ease of use, power and successful applications

9/14/05PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper  Improve several aspects of analysis  Event selection  Triggering, Real-time Filters, Data Streaming  Event reconstruction  Tracking/vertexing, particle ID  Signal/Background Discrimination  Higgs discovery, SUSY discovery, Single top, …  Functional Approximation  Jet energy corrections, tag rates, fake rates  Parameter estimation  Top quark mass, Higgs mass, SUSY model parameters  Data Exploration  Knowledge Discovery via data-mining  Data-driven extraction of information, latent structure analysis Why Multivariate Methods?

9/14/05PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper Multi Layer Perceptron  A popular and powerful neural network model: i j k  ji  kj Need to find  ’s and  ’s, the free parameters of the model

9/14/05PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper The Bayesian Connection Output of a feed forward neural network can approximate the posterior probability P(s|x 1,x 2 ).

9/14/05PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper The Top Quark Post-Evidence, Pre-Discovery ! Fisher Analysis of tt  e  channel One candidate event (S/B)(m t = 180 GeV) = 18 w.r.t. Z  = 10 w.r.t WW NN Analysis tt  e+jets channel tt W+jets tt160Data P. Bhat, DPF94

9/14/05PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper Measuring the Top Quark Mass Discriminant variables m t = ± 5.6(stat.) ± 6.2 (syst.) GeV/c 2 The Discriminants DØ Lepton+jets Fit performed in 2-D: (D LB/NN, m fit )

9/14/05PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper Higgs Discovery Reach  The challenges are daunting! But using NN provides same reach with a factor of 2 less luminosity w.r.t. conventional analysis  Improved bb mass resolution & b-tag efficiency crucial Run II Higgs study hep-ph/ (Oct-2000) P.C.Bhat, R.Gilmartin, H.Prosper, Phys.Rev.D.62 (2000)

9/14/05PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper Limitations of “Conventional NN”  The training yields one set of weights or network parameters  Need to look for “best” network, but avoid overfitting  Heuristic decisions on network architecture  Inputs, number of hidden nodes, etc.  No direct way to compute uncertainties

9/14/05PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper Ensembles of Networks NN 1 NN 2 NN 3 NN M X y1y1 y2y2 y3y3 yMyM Decision by averaging over many networks (a committee of networks) has lower error than that of any individual network.

9/14/05PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper Bayesian Learning  The result of Bayesian training is a posterior density of the network weights  P(w|training data)  Generate a sequence of weights (network parameters) in the network parameter space i.e., a sequence of networks. The optimal network is approximated by averaging over the last K points:

9/14/05PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper Bayesian Learning – 2  Advantages  Less prone to over-fitting, because of Bayesian averaging.  Less need to optimize the size of the network. Can use a large network! Indeed, number of weights can be greater than number of training events! p(t|x)  In principle, provides best estimate of p(t|x)  Disadvantages  Computationally demanding!

9/14/05PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper Bayesian Learning – 3  Computationally demanding because  The dimensionality of the parameter space is, typically, large.  There could be multiple maxima in the likelihood function p(t|x,w), or, equivalently, multiple minima in the error function E(x,w).

9/14/05PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper Bayesian Neural Networks – 1  Basic Idea  Compute  Then estimate p(t|x new ) by averaging over NNs LikelihoodPrior

9/14/05PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper Bayesian Neural Networks – 2  Likelihood  Where t i = 0 or 1 for background/signal  Prior

9/14/05PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper Bayesian Neural Networks – 3  Computational method  Generate a Markov chain (MC) of N points {w} from the posterior density p(w|x) and average over last K  Markov Chain Monte Carlo software from by Radford Neal

9/14/05PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper Bayesian Neural Networks – 4  Treat sampling of posterior density as a problem in Hamiltonian dynamics in which the phase space (p,q) is explored using Markov techniques

9/14/05PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper A Simple Example  Signal  ppbar  tqb (  channel)  Background  ppbar  Wbb  NN Model  (1, 15, 1)  MCMC  5000 tqb + Wbb events  Use last 20 networks in a MC chain of 500. HT_AllJets_MinusBestJets (scaled) Wbb tqb

9/14/05PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper A Simple Example Estimate of Prob(s|H T ) Blue dots: p(s|H T ) = H tqb /(H tqb +H Wbb ) Curves: (individual NNs) y(H T, w n ) Black curve:

9/14/05PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper Example: Single Top Search  Training Data  2000 events (1000 tqb-  Wbb-  )  Standard set of 11 variables  Network 391  (11, 30, 1) Network (391 parameters!)  Markov Chain Monte Carlo (MCMC)  500 iterations, but use last 100 iterations  20 MCMC steps per iteration  NN-parameters stored after each iteration  10,000 steps  ~ 1000 steps / hour (on 1 GHz, Pentium III laptop)

9/14/05PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper Signal/Bkgd. Distributions

9/14/05PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper

9/14/05PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper Weighting with NN output  Number of data events:  Create weighted histograms of variables

9/14/05PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper Weighted Distributions Magenta: Weighting signal only; Blue: Weighting signal & background Black: Un-weighted signal distribution

9/14/05PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper Summary  Bayesian learning of neural networks takes us another step closer to realizing optimal results in classification (or density estimation) problems. It allows a fully probabilistic approach with proper treatment of uncertainties.  We have started to explore Bayesian neural networks and the initial results are promising, though computationally challenging.