Bayesian Neural Networks

1. Bayesian Neural Networks Pushpa Bhat Fermilab Harrison Prosper Florida State University

2. Outline • Introduction • Bayesian Learning • Simple Examples • Summary PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper

3. 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 PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper

4. Why Multivariate Methods? • 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 PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper

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

6. The Bayesian Connection Output of a feed forward neural network can approximate the posterior probability P(s|x1,x2). PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper

7. The Top QuarkPost-Evidence, Pre-Discovery ! P. Bhat, DPF94 Fisher Analysis of tte channel One candidate event (S/B)(mt = 180 GeV) = 18 w.r.t. Z = 10 w.r.t WW NN Analysis tt  e+jets channel tt W+jets tt160 Data W+jets PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper

8. Measuring the Top Quark Mass DØ Lepton+jets Discriminant variables The Discriminants mt = 173.3 ± 5.6(stat.) ± 6.2 (syst.) GeV/c2 Fit performed in 2-D: (DLB/NN, mfit) PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper

9. 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/0010338 (Oct-2000) P.C.Bhat, R.Gilmartin, H.Prosper, Phys.Rev.D.62 (2000) 074022 PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper

10. 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 PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper

11. Ensembles of Networks NN1 y1 NN2 y2 X NN3 y3 NNM yM Decision by averaging over many networks (a committee of networks) has lower error than that of any individual network. PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper

12. Bayesian Learning • The result of Bayesian training is a posteriordensity 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: PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper

13. 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! • In principle, provides best estimate of p(t|x) • Disadvantages • Computationally demanding! PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper

14. 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). PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper

15. Bayesian Neural Networks – 1 • Basic Idea • Compute • Then estimate p(t|xnew) by averaging over NNs Likelihood Prior PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper

16. Bayesian Neural Networks – 2 • Likelihood • Where ti = 0 or 1 for background/signal • Prior PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper

17. 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 http://www.cs.toronto.edu/~radford/fbm.software.html by Radford Neal PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper

18. 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 PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper

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

20. A Simple Example Estimate of Prob(s|HT) Blue dots: p(s|HT) = Htqb/(Htqb+HWbb) Curves: (individual NNs) y(HT, wn) Black curve: < y(HT, w) > PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper

21. Example: Single Top Search • Training Data • 2000 events (1000 tqb-m + 1000 Wbb-m) • Standard set of 11 variables • Network • (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) PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper

22. Signal/Bkgd. Distributions PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper

23. PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper

24. Weighting with NN output • Number of data events: • Create weighted histograms of variables PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper

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

26. 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. PHYSTAT05 Oxford Bayesian Neural Networks Bhat/Prosper