1

A Bayesian statistical methodfor particle identification

in shower counters

IX International Workshop on Advanced Computing and Analysis Techniques in Physics Research December 1-5, 2003

N.Takashimizu1, A.Kimura2, A.shibata3 and T.Sasaki31 Shimane University

2 Ritsumeikan University3 High Energy Accelerator Research Organization

2

Introduction

• We made an attempt to identify particle using Bayesian statistical method.

• The particle identification will be possible by extracting pattern of showers because the energy distribution differ with incident particle or energy.

• Using Bayesian method in addition to the existing particle identification method, the improvement of experimental precision is expected.

3

Bayes’ Theorem

• Bayes’ theorem is a simple formula which gives the probability of a hypothesis H from an observation A.

• We can calculate the conditional probability of H which causes A as follows.

– P(A|H) : The probability of A given by H– P(H) : The probability prior to the observations– P(A) : The probability of A whether H is true or not

• Bayes’ theorem gives a learning system how to update parameters after observing A.

P(A|H)P(H)P(H|A)

P(A)＝

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Bayesian Estimation

• Bayesian estimation is a statistical method based on the Bayes’ theorem.– Think of unknown parameters as probability variables and gi

ve them density distributions instead of estimating particular value.

• Represent information about parameters as prior distribution p （ θ， x） before we make observations.– Generally the prior distribution is not sharp because our k

nowledge about parameter is insufficient before observation.

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Bayesian Estimation

• When we make an observation the posterior distribution can be calculated by using both data generation model and prior distribution.

• The predictive distribution of the future observation based on the observed data x= （ x1 ， x2 ，… xn ） is the expectation of the model for all possible posterior distribution.

| ( | ) ( )p x m x p

1 1| ( | ) ( | )n np x m x p d x x

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Appling to the shower

• Now we apply the bayesian estimation to the electromagretic shower

P()

m(x|)

P(|x)

P(x n+1|x)

Model of the energy deposit in the showercharacterized by mean and variance

Conditional distributionof N events given

Prediction of the next event

Prior distribution of parameters

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Shower Modeling

• Divide a calorimeter into 16 blocks vertically to the incident direction.

• Model distribution of electromagnetic shower is denoted in terms of the sum of energy deposit in each block ………… Nb (Nb= 16).

b2, ,..., N ε

1 2

Nb

x

y

z

… …

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Model Distribution

• If the shape of the shower is multivariate normal distribution N(θ,Σ) then the model is presented as

• When the shower is caused by particle with incident energy E0 the model above is represented by

• To simplify the calculation we assume there is no correlation among energy deposit in each block.

1 1| exp

22

bN

m

ε θ Σ Σ ε θ Σ ε θ

2 2 21 2diag , ,...,

bN Σ

0|m ε θ Σ

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Model Distribution

• After N observation the model will be a joint probability density

0 01

1

1

1

| |

1 1exp tr{ }

22

1

b

n

i ii

nN

kkk k

n

ii

n

i ii

m m

n S

n

where

ε θ Σ ε θ Σ

Σ ε θ ε θ

ε ε

S ε ε ε ε

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Posterior Distribution

• When we assume prior distribution is uniform, it is given by

• The posterior distribution is given in terms of the model and the prior distribution when observing n showers caused by , E0

2 20

1 1

|b bN N

k k kk k

P P P P P

θ Σ θ Σ

0 0 0| | , |P EX m P θ Σ ε θ Σ θ Σ

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Predictive Distribution

• Finally the next shower can be predicted on condition that n- shower, particle and incident energy are known.

1 0 1 0 0

21

1

2

1

1

| | |

2 / 112 2

1 1exp

2 /2 /

b

b

b

n n

N

nN

kk n

k kk kk

Nn k

k kkkk

P EX m P EX d d

nSn

S n n Sn n

S nS n

ε ε θ Σ

ε

ε ε

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Particle Identification

• Given the next shower the conditional probability for occurrence of that shower is obtained from the predictive distribution.

• Selecting the most probable condition, that is, a parameter set of and E0, enable us the particle identification.

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Bayesian Learning for simulation data

• Monte Carlo simulation(Geant4)– Calorimeter configuration

• Material : Lead Grass Pb (66.0%), O (19.9%), Si (12.7%), K (0.8%), Na (0.4%), As (0.2%) density ： 5.2 g/cm3

• Size : 20cm• Structure : A total of 20*20*20 lead grass of 1cm cube

x

yz

20

20

20

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Bayesian Learning for simulation data

– Incident angle : (0,0,1)

– Incident position : (10,10,0)– Data for learning :

= (e-,-) E0 = (0.5,1.0,2.0,3.0)GeV

Incident directionIncident direction

y

x

z

Incident directionIncident direction

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Energy distribution

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Result

e －， 0.5e －， 1.0 －， 0.5e －， 3.0e －， 2.0 －， 1.0 －， 2.0 －， 3.0

e －， 0.5

e －， 1.0

e －， 2.0

e －， 3.0

－， 0.5

－， 1.0

－， 2.0 －， 3.0

801 108 2 29

06 0 0 0

009

2419

44 8972000100

653

8940

47300

860

953210

37

0004

2937

3

000

948849849

000

144634

140000

195161

Condition

Dat

a fo

r le

arni

ng

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Summary

• We made an attempt to identify particle by means of modeling the shower profile based on Bayesian statistics and develop the possibility for Bayesian approach.

• Without any other information e.g. charges of particles given by tracking detectors, we have obtained a high percentage of correct identification for e and

• Future plan

• improvement of model and prior distribution