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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
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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.
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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