Quasi-optimal observables: Attaining the quality of maximal likelihood in parameter estimation when only a MC event generator is available

In this lecture, I’d like to explain a recent finding [1] which connects the two basic methods of parameter estimation, the method of maximal likelihood and the method of generalized moments (see e.g. [2]). The two methods (along with the χ method, which I won’t discuss) are very well known and widely used in experimental physics. In a sense, the connection views the method of maximal likelihood as corresponding to a special point in the space of generalized moments, and considers small deviations from that point. The point corresponds to the minimum of the fundamental Cramer-Rao inequality, and small deviations from it introduce non-optimalities (compared with the maximal likelihood method) that are only quadratic in the deviations. This approach offers what appears to be a new and useful algorithmic scheme which combines the theoretical advantage of the method of maximal likelihood (i.e. the fact that it yields the absolute minimum for the variance of the parameter being estimated with a given data sample) with the algorithmic simplicity of the method of moments. I call the resulting method the method of quasi-optimal observables. It is useful in situations where the method of maximal likelihood fails or cannot be applied, e.g. in high energy physics where typically only a Monte Carlo event generator is available but no explicit formula for the probability density. One deals with a random variable P whose instances (specific values) are called events. Their probability density is denoted as π(P). It is assumed to depend on a parameter M which has to be estimated from an experimental sample of events {Pi}i . The method of generalized moments consists in choosing a function f (P) defined on events (the generalized moment or, using the language of quantum theory, observable), and then finding M by fitting its theoretical average value,