Bayes at the Frontier: The Promise and Challenges

Transcription

1 : The Promise and Department of Physics, Florida State University September 21, 2012

2 Outline Introduction 1 Introduction

3 21st century astrophysics, cosmology, and high energy physics share many characteristics: Each uses well-defined physical theories Each must deal with uncertainty Each must deal, or will have to deal, with Big Data Each must solve the Is this real? problem

4 ...and, from time to time, each makes a spectacular discovery : Accelerating expansion 2012: Higgs-like particle distance modulus µ The Union2.1 Compilation The Supernova Cosmology Project 35 SCPUnion2.1_mu_vs_z.txt redshift z Events / ( 1 GeV ) 2400 Data CMS Preliminary 2200 s = 7 TeV, L = 5.1 fb S+B Fit Bkg Fit Component 2000 s = 8 TeV, L = 5.3 fb ±1 σ 1800 ±2 σ All Categories Combined m γγ (GeV)

5 ...and, we even have pictures! 1995: DØ Top Quark Discovery Data

6 High Energy Physics What is high energy physics? The study of matter under extreme conditions. Temperatures billion times hotter than the solar core! How do we do this? Create trillions of collisions between particles (called events) For each selected event, measure properties of particle debris Make predictions through simulation of billions of events Compare data with predictions and cull models that fail

7 Outline Introduction 1 Introduction

8 LHCb ATLAS CMS ALICE Courtesy CERN

9 The CMS Experiment

10 Introduction The ATLAS Experiment

11 Physics Papers Timeline CMS The Experiment Large Hadron Collider...papers... Show all Total QCD Physics Exotica Searches Supersymmetry B Physics Electroweak Top Physics Heavy Ion Higgs Forward Physics Standard Model Search for three-jet resonances in pp Search for a W' boson decaying... Search for pair production of first-... Search for heavy Majorana neutrinos in... Search for new physics with long-lived... Search for stopped long-lived particles produced Search for dark matter and large... Search for charge-asymmetric production of W'... Search for narrow resonances in dilepton... Search for high mass resonances decaying... Search for exotic particles decaying to Search for heavy long-lived charged particles... Search for leptonic decays of W'... Search for anomalous $t \bar{t}$ production... Search for heavy bottom-like quarks in... Search for Dark Matter and Large Search for heavy, top-like quark pair... Search for microscopic black holes in... Search for quark compositeness in dijet... Search for large extra dimensions in... Search for signatures of extra dimensions Search for a Vectorlike Quark with... Search for Resonances in the Dijet... Search for Three-Jet Resonances in pp... A search for excited leptons in... Search for New Physics with a Search for Light Resonances Decaying into... Search for Same-Sign Top-Quark Pair Production... Search for First Generation Scalar Leptoquarks... Search for Large Extra Dimensions in... Search for Resonances in the Dilepton Search for a W' boson decaying... Search for a Heavy Bottom-like Quark... Search for Heavy Stable Charged Particles... Search for a heavy gauge boson... Search for Pair Production of First-Generation... 5 Search for Pair Production of Second-Generation... Search for Microscopic Black Hole Signatures... Search for Stopped Gluinos in pp... Search for Quark Compositeness with the... Search for Dijet Resonances in Jan 2010 Apr 2010 Jul 2010 Oct 2010 Jan 2011 Apr 2011 Jul 2011 Oct 2011 Jan 2012 Apr 2012 Jul 2012

12 Physics Papers Timeline CMS Experiment Introduction Show all Total QCD Physics Exotica Searches Supersymmetry B Physics Electroweak Top Physics Heavy Ion Higgs Forward Physics Standard Model...and yet more papers Observation of a new boson at... A search for a doubly-charged Higgs Search for a fermiophobic Higgs boson... Search for a light pseudoscalar Higgs Search for a light charged Higgs... Search for the standard model Higgs Search for neutral Higgs bosons decaying... Search for the standard model Higgs... 8 Search for the standard model Higgs... Search for the standard model Higgs... 6 Combined results of searches for the... Search for a Higgs boson in... 4 Search for the standard model Higgs... Search for the standard model Higgs... 2 Search for Neutral Minimal Supersymmetric Standard... Measurement of $W^+W^-$ Production and Search... 0 Jan 2010 Apr 2010 Jul 2010 Oct 2010 Jan 2011 Apr 2011 Jul 2011 Oct 2011 Jan 2012 Apr 2012 Jul 2012

13 Outline Introduction 1 Introduction

14 In 1961, Glashow created a theory that unified electromagnetism and the weak nuclear force. But his theory described a world of massless particles. In 1964, Brout, Englert, Guralnik, Hagen, Higgs, and Kibble answered the question: how does the mass of elementary particles, such as the electron, arise? In 1967, their idea was incorporated into the electroweak theory of Glashow, Salam, and Weinberg, which became part of the Standard Model (SM). In 2012, ATLAS and CMS announce the discovery of a Higgs-like particle with a mass 125 GeV.

15 The SM predicts everything about the Higgs boson except its mass. In particular, it predicts that this particle will decay rapidly in one of many ways, including: H γγ H ZZ 4l In 2011 and 2012, the LHC produced 800 trillion collisions with a data rate of 40 Tbytes per second of which 400 Mbyte per second were selected and stored yielding 15 Pbyte per year and 50, 000 γγ events after final filtering

16 A γγ Event from CMS

17 Spectral Model Di-photon (γγ) spectrum modeled with the function n f (x; s, m, w, a) = a r x r + µ F s (x; m, w) r=0 x = m γγ is the di-photon mass, a a r, r = 0... n are the background nuisance parameters, F s = signal distribution ( F s (x;...) dx = SM prediction) µ = the expected signal / SM prediction, m = mass of the particle, w = width of signal.

18 CMS: 7 TeV p + p γγ data Events / ( 1 GeV ) 20 CMS Preliminary Data S+B Fit s = 7 TeV, L = 5.1 fb 18 Bkg Fit Component ±1 σ 16 ±2 σ BDT >= m γγ (GeV) Events / ( 1 GeV ) 250 CMS Preliminary Data S+B Fit s = 7 TeV, L = 5.1 fb Bkg Fit Component ±1 σ ±2 σ 0.55 <= BDT < m γγ (GeV) Events / ( 1 GeV ) 10 CMS Preliminary Data S+B Fit s = 7 TeV, L = 5.1 fb Bkg Fit Component ±1 σ ±2 σ BDT >= 0.05 Di-jet Tag m γγ (GeV) Events / ( 1 GeV ) CMS Preliminary s = 7 TeV, L = 5.1 fb Data S+B Fit Bkg Fit Component ±1 σ ±2 σ 0.74 <= BDT < 0.89 Events / ( 1 GeV ) CMS Preliminary s = 7 TeV, L = 5.1 fb Data S+B Fit Bkg Fit Component ±1 σ ±2 σ 0.05 <= BDT < m γγ (GeV) m γγ (GeV)

19 CMS: 8 TeV p + p γγ data Events / ( 1 GeV ) CMS Preliminary s = 8 TeV, L = 5.3 fb Data S+B Fit Bkg Fit Component ±1 σ ±2 σ BDT >= m γγ (GeV) Events / ( 1 GeV ) CMS Preliminary s = 8 TeV, L = 5.3 fb Data S+B Fit Bkg Fit Component ±1 σ ±2 σ 0.50 <= BDT < m γγ (GeV) Events / ( 1 GeV ) CMS Preliminary s = 8 TeV, L = 5.3 fb Data S+B Fit Bkg Fit Component ±1 σ ±2 σ BDT >= Tight Di-jet Tag m γγ (GeV) Events / ( 1 GeV ) CMS Preliminary s = 8 TeV, L = 5.3 fb Data S+B Fit Bkg Fit Component ±1 σ ±2 σ 0.71 <= BDT < 0.88 Events / ( 1 GeV ) CMS Preliminary s = 8 TeV, L = 5.3 fb Data S+B Fit Bkg Fit Component ±1 σ ±2 σ <= BDT < m γγ (GeV) m γγ (GeV)

20 p-value vs Higgs boson mass Local p-value Interpretation Requires LEE CMS Preliminary s = 7 TeV, L = 5.1 fb s = 8 TeV, L = 5.3 fb 1σ 2σ Observed (Asymptotic) 1x SM Higgs Expected (Asimov) 3σ TeV Observed (Asymptotic) 8 TeV Observed (Asymptotic) m H (GeV) 4σ

21 Outline Introduction 1 Introduction

22

23 If quarks have substructure, we expect to see deviations in the momentum spectrum of jets from the prediction of the SM. ) dη (GeV N/dp 2 d T CMS Preliminary L = 5 fb Data Λ = 8 TeV Λ = 10 TeV Λ = 12 TeV Λ = 14 TeV QCD s = 7 TeV NLO Data / QCD CMS Preliminary 3 2 Data Λ = 8 TeV Λ = 10 TeV Λ = 12 TeV Λ = 14 TeV L = 5 fb s = 7 TeV destructive interference Jet p (GeV) T 1 destructive interference Jet p (GeV) T Deviations characterized with a single parameter λ. The expected count in the k th bin is given by n k = α σ k, where α is a normalization factor and σ k = c k + b k λ + a k λ 2 is the cross section.

24 Bayesian analysis Step 1: Likelihood p(d λ, α, ω) = K Poisson(N k α σ k ), k=1 = exp( α σ) α N K = Poisson(N ασ) k=1 σ N k k /N k!, Multinomial(N 1,, N K ; σ 1 σ,, σ K ), (1) σ where σ K k=1 σ k and N K k=1 N k is the total observed count.

25 Step 2: Prior Write π(α, λ, ω) = π(α λ, ω)π(λ, ω), (2) Model π(α λ, ω) with a gamma density and marginalize with respect to α. This yields p(d λ, ω) (1 z) 2 Beta(z; N + 1, Q) Multinomial(N 1,, N K ; σ 1 σ,, σ K ), (3) σ where z σ/(σ + q) and q and Q are known constants.

26 Step 3: Posterior Approximate marginalization with respect to ω by averaging p(d λ, ω) over a sample of points ω i. Finally, compute the Jeffreys prior for p(d λ) and compute the posterior density p(λ D).

27 Outline Introduction 1 Introduction

28 Sociological challenges: Coverage is king! Give me p-values or give me death! Computational challenges: Extending to multi-parameter problems (e.g., SUSY) Bernardo/Berger recursive reference prior algorithm? In principle, yes. But, in practice, everything has to be done computationally. Question: Is there a way to sample parameter points so that the resulting swarm of points approximates an n-dimensional reference prior?

29 The Minimal Supersymmetric Standard Model

30 Outline Introduction 1 Introduction

31 Frequentist methods are still the norm in high energy physics However, Bayesian methods have made some inroads in some high-profile analyses The main difficulty is the absence of off-the-shelf computational methods for constructing multi-dimensional (reference) priors Sparsity is important in high energy physics. Traditionally, this is done by hand. But, perhaps we are now moving into an era in which this could be automated?

32 Acknowledgements Thanks to the organizers for the invitation Many thanks to my ( 10,000) colleagues at CERN and around the world Many thanks to the taxpayers of the world for their very generous support