Concluding Talk: Physics

The general problem that we wish to solve is given a measurement of an event X = (x1, x2, . . . xn), find the function F(X) which returns 1 if the event is signal (s) and 0 if the event is background (b) to optimize a figure of merit, say s /√ b for a discovery or s /√ s + b for an established signal. In principle the solution is straightforward. Use a Monte Carlo simulation to calculate the likelihood ratio Ls(X)/Lb(X) and derive F(X) from it. This just amounts to counting the number of signal and background at each point in the parameter space. Then F can be calculated by ordering the likelihood ratios and accepting them in decreasing order until the figure of merit starts to decrease. Further, by the Neyman-Pearson Theorem, this is the optimum solution. Of course, this does not work due to the “curse of dimensionality.” In a high-dimension space, even the largest data set is sparse, with the distance between neighboring events comparable to the radius of the space. Thus, we are forced to substitute cleverness for brute force. In recent years, physicists have come to learn that computers may be cleverer than they are; they have turned to machine learning. One gives the computer samples of signal and background events and lets the computer figure out what F(X) is. 2.2. Artificial Neural Networks