signal/background discrimination harrison b. prosper samsi, march 20061 signal/background...

Signal/Background Discrimination Harrison B. Prosper SAMSI, March 2006 1

Signal/Background Signal/Background Discrimination Discrimination

in in Particle PhysicsParticle Physics

Harrison B. ProsperFlorida State University

SAMSI8 March, 2006


OutlineOutline

Particle Physics Data

Signal/Background Discrimination

Summary


Particle Physics DataParticle Physics Data

proton + anti-proton

-> positron (e+)neutrino ()Jet1Jet2Jet3Jet4

This event is described by(at least) 3 + 2 + 3 x 4 = 17measured quantities.


Particle Physics DataParticle Physics Data

H0 Standard ModelH1 Model of the Week

1

106


Signal/Background Signal/Background DiscriminationDiscrimination

To minimize misclassification probability, compute

p(S|x) = p(x|S) p(S) / [p(x|S) p(S) + p(x|B) p(B)]

Every signal/background discrimination method is ultimately an algorithm to approximate this function, or a mapping thereof.

p(s) / p(b) is the prior signal to background ratio, that is, it is S/B before applying a cut to p(S|x).


GivenD D = x, y

x = {x1,…xN}, y = {y1,…yN}of N training examples (events)

Infer A discriminant function f(x, w), with parameters w p(ww|x, y) = p(x, y|ww) p(ww) / p(x, y)

= p(y|x, w) p(x|ww) p(ww) / p(y|x) p(x)= p(y|x, w) p(ww) / p(y|x)

assuming p(x|w) -> p(x)



A typical likelihood for classification:

p(y|x, ww) = i f(xi, ww)y [1 – f(xi, ww)]1-y

where y = 0 for background eventsy = 1 for signal events

If f(x, ww) flexible enough, then maximizing p(y|x, ww) with respect to w yields f = p(S|x), asymptotically.



However, in a Bayesian calculation it is more natural to average

y(x) = ∫ f(x, ww) p(ww|D) dw

Questions:1. Do suitably flexible functions f(x, ww) exist?

2. Is there a feasible way to do the integral?



Answer 1: Yes!Answer 1: Yes!

Hilbert’s 13th problem: Prove a special case of the conjecture:The following is impossible, in general,

f(x1,…,xn) = F( g1(x1),…, gn(xn) )

In 1957, Kolmogorov proved thecontrary: A function f:Rn -> R can berepresented as followsf(x1,..,xn) = ∑i=1

2n+1 Qi( ∑j=1n Gij(xj) )

where Gij are independent of f(.)


Kolmogorov FunctionsKolmogorov Functions

H

j

P

iiijjj xuavbwxf

1 1

tanh),(

n(x,w)

x1

x2

u, a

v, b )],(exp[1

1),(

wxfwxn

A neural network is an example of a Kolmogorov function, that is, a function capable of approximating arbitrary mappings f:Rn -> R

The parameters w = (u, a, v, b) are called weightsweights


Answer 2: Yes!Answer 2: Yes!

Computational MethodGenerate a Markov chain (MC) of N points

{w}, whose stationary density is p(w|D), and average over the last M points.

Map problem into that of “particle” moving in a spatially-varying “potential” and use methods of statistical mechanics to generate states (p, w) with probability

~ exp(- H),

where H is the “Hamiltonian”H = log p(w|D) + p2, with

“momentum” p.


Hybrid Markov Chain Monte Hybrid Markov Chain Monte CarloCarlo

Computational Method…For a fixed H traverse space (p, w) using

Hamilton’s equations, which guarantees that all points consistent with H will be visited with equal probability ~ exp(-H).

To allow exploration of states with differing values of H one introduces, periodically, random changes to the momentum p.

SoftwareFlexible Bayesian Modeling by Radford Nealhttp://www.cs.utoronto.ca/~radford/fbm.software.html

Example 1Example 1


Example 1: 1-DExample 1: 1-D

Signal p+pbar -> t q b

Background p+pbar -> W b b

NN Model Class (1, 15, 1)

MCMC 500 tqb + Wbb events Use last 20 points in a

chain of 10,000,

x

tqb

skipping every 20th

Wbb


Example 1: 1-DExample 1: 1-D

x

Dots p(S|x) = HS/(HS+HB)

HS, HB, 1-D histograms

Curves Individual NNs n(x, wwkk)

Black curve < n(x, w) >

Example 2Example 2


Example 2: 14-D (Finding Example 2: 14-D (Finding Susy!)Susy!)

Transversemomentumspectra

Signal:blackcurve

Signal/Noise

1/25,0001/25,000



Missingtransversemomentumspectrum

(caused byescape ofneutrinosand Susyparticles)

Measuredquantities:

4 x (ET, , )

+ (ET, )

= 1414


Likelihood Prior


Signal250 p+pbar -> gluino, gluino (Susy) events

Background250 p+pbar -> top, anti-top events

NN Model Class(14, 40, 1) (w є 641-D parameter space!)

MCMCUse last 100 networks in a Markov chain of

10,000, skipping every 20.


ResultsResults

Network distributionbeyond n(x) > 0.9

Assuming L = 10 fb-1

Cut S B S/√B0.90 5x103 2x106 3.50.95 4x103 7x105 4.70.99 1x103 2x104 7.0


But Does It Really Work? But Does It Really Work?

Let d(x) = N p(x|S) + N p(x|B) be the density of the data, containing 2N events, assuming, for simplicity, p(S) = p(B).

A properly trained classifier y(x) approximates

p(S|x) = p(x|S)/[p(x|S) + p(x|B)]

Therefore, if the data (signal + background) are weighted with y(x), we should recover the signal density.


But Does It Really Work? But Does It Really Work?

It seems to!

Example 3Example 3


Particle Physics Data, Take 2Particle Physics Data, Take 2

Two varieties of jet:

1. Tagged (Jet 1, Jet 4)

2. Untagged (Jet 2, Jet 3)

We are often interested in

Pr(Tagged|Jet Variables)


Probability Density Probability Density EstimationEstimation

Approximate a density by a sum over kernels K(.), one placed at each of the N points xi of the training sample.

h is one or more smoothing parameters adjusted to provide the best approximation to the true density p(x).

If h is too small, the model will be very spiky; if h is too large, features of the density p(x) will be lost.

N

i

i

h

xxk

Nxp

1

1)(ˆ



Why does this work? Consider the limit as N -> ∞ of

In the limit N -> ∞, the true density p(x) will be recovered provided that h -> 0 in such a way that

N

i

i

h

xxk

Nxp

1

1)(ˆ

dzzp

h

zxkxp )(

)()(ˆ

)( zxh

xxk ni



As long as the kernel behaves sensibly in the N -> ∞ limit any kernel will do. In practice, the most commonly used kernel is the product of 1-D Gaussians, one for each dimension “i”:

One advantage of the PDE approximation is that it contains very few adjustable parameters: basically, the smoothing parameters.

2/

1

2

)2(/2/exp/ nnn

i

i hh

zxhzxk



Tagged-jet


Projections of estimated p(T|x) (black curve) onto thePT, and axes. Blue points: ratio of blue to red histograms

(see slide 25)



Tagged-jet


Projections of data weighted by p(T|x). Recovers tagged density p(xx|T).


But, How Well Does It Work?But, How Well Does It Work?

Tagged-jet


How well do the n-D model and the n-D data agree?A thought (JL, HBP):

1. Project the model and the data onto the same set ofrandomly directed rays through the origin.2. Compute some measure of discrepancy for each pairof projections.3. Do something sensible with this set of numbers!!


Tagged-jet



Projections of p(T|x)onto 3 randomly chosenrays through the origin.


Tagged-jet


Projections of weighted tagged + untagged data onto the 3 randomly selected rays.



SummarySummary

Multivariate methods have been applied with considerable success in particle physics, especially for classification. However, there is considerable room for improving our understanding of them as well as expanding their domain of application.

The main challenge is data/model comparison when each datum is a point in 1…20 dimensions. During the SAMSI workshop we hope to make some progress on the use of projections onto multiple rays. This may be an interesting area for collaboration between physicists and statisticians.

signal/background discrimination harrison b. prosper samsi, march 20061 signal/background...

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