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

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.

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.

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.

Results: xy Event Event Blue: 1…500 Red: 501…1000

Results:  Event Event Blue: 1…500 Red: 501…1000

Results: E Event Event Blue: 1…1000 Red: 501…1000

Results: x Event Event Blue: 1…1000 Red: 501…1000

Results: y Event Event Blue: 1…1000 Red: 501…1000

Results: z Event Event Blue: 1…1000 Red: 501…1000

Results:  Event Event Blue: 1…1000 Red: 501…1000

Results:  Event Event Blue: 1…1000 Red: 501…1000

Summary Event Event x = y = z = th = ph = E = % x = y = z = th = ph = E = %

Simulation at best point vs. data Event Event

Simulation at best point vs. data Event Event

Simulation at best point vs. data Event Event

Simulation at best point vs. data Event Event

Simulation at best point vs. data Event Event

Simulation at best point vs. data Event Event

Simulation at best point vs. data Event Event

Simulation at best point vs. data Event Event

Simulation at best point vs. data Event Event

Simulation at best point vs. data Event Event

Simulation at best point vs. data Event Event

Simulation at best point vs. data Event Event

Simulation at best point vs. data Event Event

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