1. 1 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.Takashimizu 1, A.Kimura 2, A.shibata 3 and T.Sasaki 3 1 Shimane University 2 Ritsumeikan University 3 High Energy Accelerator Research Organization

2. 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. 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.

4. 4 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.

5. 5 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= （ x 1 ， x 2 ， …x n ） is the expectation of the model for all possible posterior distribution.

6. 6 Appling to the shower Now we apply the bayesian estimation to the electromagretic shower P(  ) m(x| q ) P(  |x) P(x n+1 |x) Model of the energy deposit in the shower characterized by mean  and variance  Conditional distribution  of N events given Prediction of the next event Prior distribution of parameters

7. 7 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,…………..,  Nb (Nb= 16). 11 22  Nb x y z ……

8. 8 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 E 0 the model above is represented by To simplify the calculation we assume there is no correlation among energy deposit in each block.

9. 9 Model Distribution After N observation the model will be a joint probability density

10. 10 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 , E 0

11. 11 Predictive Distribution Finally the next shower can be predicted on condition that n- shower, particle and incident energy are known.

12. 12 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 E 0, enable us the particle identification.

13. 13 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/cm 3 Size : 20cm Structure : A total of 20*20*20 lead grass of 1cm cube x y z 20

14. 14 Bayesian Learning for simulation data –Incident angle : (0,0,1) –Incident position : (10,10,0) –Data for learning :  = (e-,  -) E 0 = (0.5,1.0,2.0,3.0)GeV Incident direction y x z

15. 15 Energy distribution

16. 16 Result e － ， 0.5e － ， 1.0  － ， 0.5 e － ， 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  － ， 3.0 801 108 229 0 600 0 0 0 9 24 19 44897 20 0 0 1 0 0 6 53 894 0 47 3 0 0 86 0 953 2 1 0 37 0 0 0 4 29 37 3 0 0 0 948 849 0 0 0 14 46 34 14 0 0 0 0 19 51 61 Condition Data for learning

17. 17 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