1. MCMC reconstruction of the 2 HE cascade events Dmitry Chirkin, UW Madison

2. Markov Chain Monte Carlo 1.Start with some initial values, e.g., x,y,z=COG; ,  =0,0; E=1.e5, t=0 or even x,y,z=0,0,0 (as in the example on first slide). 2.Simulate EM cascade with these parameters (point 1). 3.Compute the likelihood L 1 quantifying the difference between this simulation and data (here: same as in SPICE fits, or can be of your choice, e.g.,  2 ). 4.Sample the next point 2 from a proposal distribution, e.g., gaussian centered on the cascade parameters at point 1. 5.Calculate the likelihood L 2 at new point 2. If L 2 >L 1 jump to the new point 2, otherwise stay at 1. 6.Repeat from step 2 until chain converges to stationary state.

3. Configuration choices Simulation: ppc with SPICE Lea Likelihood: as in SPICE fit depends on number of data events n d and simulated events n s combine 25 ns. time bins with bayesian blocks algorithm Start with COG; calculate best time offset t and scale energy to maximize likelihood after each simulation (this reduces the number of parameters that are varied in MCMC). The energy scaling is achieved by fitting for n s.

4. Uncertainties and systematics The result of MCMC is a set of points that are distributed near the best fit set of parameters. The spread of MCMC points characterizes the parameter uncertainties. This is how to include systematic uncertainties, e.g., ice model: before running the simulation at points 1 and 2 first pick the model with pre-determined probabilities, e.g., 60% Spice Lea and 40% WHAM or sample directly from the error ellipse in a and e. Only statistical uncertainties are presented here.

5. Results: xy Event 118545Event 119316 Blue: 1…500 Red: 501…1000

6. Results:  Event 118545Event 119316 Blue: 1…500 Red: 501…1000

7. Results: E Event 118545Event 119316 Blue: 1…1000 Red: 501…1000

8. Results: x Event 118545Event 119316 Blue: 1…1000 Red: 501…1000

9. Results: y Event 118545Event 119316 Blue: 1…1000 Red: 501…1000

10. Results: z Event 118545Event 119316 Blue: 1…1000 Red: 501…1000

11. Results:  Event 118545Event 119316 Blue: 1…1000 Red: 501…1000

12. Results:  Event 118545Event 119316 Blue: 1…1000 Red: 501…1000

13. Summary Event 118545Event 119316 x = -202.7 +- 3.4 y = 95.8 +-3.4 z = 121.1 +- 3.9 th = 75.1+-14.3 ph = -46.0 +- 15.0 E = 1093956 +- 2.5% x = -75.2 +- 5.3 y = 261.4 +- 4.6 z = 24.4 +- 5.4 th = 31.3 +- 12.5 ph = -57.7 +- 40.2 E = 1342765 +- 9.4%

14. Simulation at best point vs. data Event 118545Event 119316

15. Simulation at best point vs. data Event 118545Event 119316

16. Simulation at best point vs. data Event 118545Event 119316

17. Simulation at best point vs. data Event 118545Event 119316

18. Simulation at best point vs. data Event 118545Event 119316

19. Simulation at best point vs. data Event 118545Event 119316

20. Simulation at best point vs. data Event 118545Event 119316

21. Simulation at best point vs. data Event 118545Event 119316

22. Simulation at best point vs. data Event 118545Event 119316

23. Simulation at best point vs. data Event 118545Event 119316

24. Simulation at best point vs. data Event 118545Event 119316

25. Simulation at best point vs. data Event 118545Event 119316

26. Simulation at best point vs. data Event 118545Event 119316

27. Concluding remarks These results are preliminary. Will run for other ice models. The likelihood value at minimum can be used to rank ice models. More plots at http://icecube.wisc.edu/~dima/work/IceCube-ftp/mcmc/.http://icecube.wisc.edu/~dima/work/IceCube-ftp/mcmc/