A Bayesian statistical method for particle identification in shower counters

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.Takashimizu1, A.Kimura2, A.shibata3 and T.Sasaki3 1 Shimane University 2 Ritsumeikan University 3 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.

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. 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.

6. 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 q and variance S m(x|q) P(q) Prior distribution of parameters Conditional distributionof N events given P(q|x) Prediction of the next event P(x n+1|x)

7. ε1 ε2 εNb 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 e１, e２,…………..,eNb(Nb= 16). … … y z x

8. Model Distribution • If the shape of the showereis multivariate normal distributionN(θ,Σ) then the model is presented as • When the shower is caused by particle f 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.

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

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 f, E0

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

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 f and E0, enable us the particle identification.

13. y z x 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 20 20

14. Incident direction x Incident direction z y Bayesian Learning for simulation data • Incident angle : (0,0,1) • Incident position : (10,10,0) • Data for learning : f = (e-,p-) E0 = (0.5,1.0,2.0,3.0)GeV

15. Energy distribution

16. Result Condition p－，1.0 p－，2.0 p－，0.5 e－，0.5 e－，1.0 e－，2.0 e－，3.0 p－，3.0 e－，0.5 108 6 2 37 3 29 14 801 e－，1.0 44 897 53 6 0 0 0 0 e－，2.0 0 20 894 86 0 0 0 0 e－，3.0 0 0 0 0 0 0 0 0 Data for learning 0 0 47 953 0 0 0 0 p－，0.5 p－，1.0 9 1 3 2 4 948 14 19 24 0 0 1 29 849 46 51 p－，2.0 p－，3.0 19 0 0 0 37 849 34 61

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 p- • Future plan • improvement of model and prior distribution