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A Bayesian statistical method
for 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.Sasaki3
1 Shimane University
2 Ritsumeikan University
3 High Energy Accelerator Research Organization
1
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.
2
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(H|A)＝
P(A|H)P(H)
P(A)
– 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.
3
Bayesian Estimation
• Bayesian estimation is a statistical method based on the
Bayes’ theorem.
– Think of unknown parameters as probability variables and
give 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
knowledge about parameter is insufficient before
observation.
4
Bayesian Estimation
• When we make an observation the posterior distribution
can be calculated by using both data generation model and
prior distribution.
p  | x   m( x |  ) p( )
• 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  xn 1 | x    m( xn 1 |  ) p( | x )d
5
Appling to the shower
•
Now we apply the bayesian estimation to the
electromagretic shower
Model of the energy deposit in the shower
characterized by mean  and variance S
m(x|q)
P()
P(|x)
P(x n+1|x)
Prior distribution of parameters
Conditional distribution of N events given
Prediction of the next event
6
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 e １ , e
２,..,e Nb (Nb= 16).

ε  e , e 2 ,..., e Nb

1
2
… …
y
z
Nb
x
7
Model Distribution
• If the shape of the shower e is multivariate normal
distribution N(θ,Σ) then the model is presented as
 1 
m  ε | θ Σ   

 2 
Nb
Σ



 1

exp   ε  θ  Σ   ε  θ 
 2

• When the shower is caused by particle f with incident energy
E0 the model above is represented by
m  ε | θ Σ  f   0 
• To simplify the calculation we assume there is no correlation
among energy deposit in each block.

Σ  diag  12 ,  22 ,...,  N2 b

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Model Distribution
• After N observation the model will be a joint probability
density
n
m  εi | θ  Σ  f   0   m  εi | θ  Σ  f   0 
i 1
 Nb
1
  
 k 1 2  k
wher e
n

 1
  
 
 exp   tr{ Σ  n   ε  θ  ε  θ   S kk  }
 
 
 2

1 n
ε   εi
n i 1
n
S    εi  ε  εi  ε 
i 1
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Posterior Distribution
• When we assume prior distribution is uniform, it is given by
Nb
   
P  θ Σ | f   0   P  θ  P  Σ   P  k  P 
k 1
2
k
Nb
k 1
2
k
• The posterior distribution is given in terms of the model and
the prior distribution when observing n showers caused by f,
E0
P  θ Σ | EX  f   0   m  ε | θ Σ , f   0  P  θ Σ | f   0 
10
Predictive Distribution
• Finally the next shower can be predicted on condition that nshower, particle and incident energy are known.
P  εn 1 | EX  f   0    m  εn 1 | f   0  P  θ Σ | EX  f   0  d d S

n 


  
  

 n  n 1  

 
 2  2 
Nb

k 1
Nb
Nb

k 1

n  Skk  εn 1  


2 Skk / n n  1  S kk 

n
2
2


ε

ε


1
1 n 1
k

exp  

2 Skk / n 
2 Skk / n


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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 f and E0, enable us the particle
identification.
12
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
20
y
z
x
20
20
13
Bayesian Learning for simulation data
– Incident angle : (0,0,1)
– Incident position : (10,10,0)
– Data for learning :
f = (e-,-) E0 = (0.5,1.0,2.0,3.0)GeV
Incident direction
x
Incident direction
y
z
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Energy distribution
15
Result
Data for learning
e－，0.5 e－，1.0 e－，2.0
e－，3.0 －，0.5 －，1.0 －，2.0 －，3.0
Condition
e－，0.5 801
e－，1.0
44
e－，2.0
0
e－，3.0
0
－，0.5
－，1.0
－，2.0
－，3.0
0
9
24
19
108
897
20
0
0
1
0
0
6
53
894
0
47
3
0
0
2
6
86
0
953
2
1
0
37
0
0
0
0
4
29
37
3
0
0
0
0
948
849
849
29
0
0
0
0
14
46
34
14
0
0
0
0
19
51
61
16
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
17