Weight optimization in multichannel

We discuss the improvement in the accuracy of a Monte Carlo integration that can be obtained by optimization of the a-priori weights of the various channels. These channels may be either the strata in a stratified-sampling approach, or the several ‘approximate’ distributions such as are used in event generators for particle phenomenology. The optimization algorithm does not require any initialization, and each Monte Carlo integration point can be used in the evaluation of the integral. We describe our experience with this method in a realistic problem, where an effective increase in program speed by almost an order of magnitude is observed. This research has been partly supported by EU under contract number CHRX-CT-0004. e-mail: [email protected] e-mail: [email protected] In almost all Monte Carlo integrations, an effort must be made to reduce the variance of the integrand [1, 2]. One of the currently popular approaches of variance reduction is the so-called stratified sampling technique, where the integration region is divided in a number of bins, with a (usually) predetermined number of integration points in each bin. An example of this technique is the program VEGAS [4], in which the bins are automatically redefined from time to time so as to reduce the integration error. Another approach is that of importance sampling , where various techniques are used to obtain (pseudo-)random variables that have a non-uniform rather than a uniform distribution: one tries to generate a density of Monte Carlo points that is larger in those parts of the integration region where also the integrand is large, thus reducing the error (note that this will work only when the integrand has large positive values; large negative values do not lend themselves to probabilistic modelling). Importance sampling is widely used in event generators for particle phenomenology: also, the Metropolis algorithm [5] used in statistical physics is actually a form of importance sampling. In the construction of event generators for particle phenomenology, the aim is usually to generate Monte Carlo points in some phase space of final-state momenta (and spins), with a density proportional to a predetermined multidifferential cross section. Often, such a cross section exhibits, in different regions of phase space, peaks that find their best description in terms of different sets of phase space variables. An example is provided by bremsstrahlung in a particle collision process, where the bremsstrahlung quanta are emitted, in the different Feynman diagrams, by different particles. It is customary, in such a case, to generate each peaking structure with a different mapping of (pseudo-)random numbers: the particular mapping used to generate an event is then chosen randomly, using a predtermined set of probabilities, which we shall call a-priori weights. It is the aim of this paper to indicate how these a-priori weights lend themselves to optimization. First, we establish some notation. The function to be integrated is f(~x), where ~x denotes a set of phase-space variables. Each distinct mapping of random numbers into ~x is called a channel , and each channel gives rise to a different (non-uniform) probability density, that we denote by gi(~x), i = 1, 2, . . . , n; n is the number of channels in our multichannel Monte Carlo. Each density gi is of course nonegative and normalized to unity: ∫ gi(~x)d~x = 1. The a-priori weights are denoted by αi, and also these must be a partition of unity: αi ≥ 0 and ∑n i=1 αi = 1. If the channels are picked at random, with probability αi for channel i, the total probability density of the obtained sample of ~x values is g(~x) = ∑n i=1 αigi(~x), which is also nonegative and normalized to unity. Note that we may take the gi(~x) to be linearly independent. The weight assigned to each Monte Carlo point must, then, be w(~x) = f(~x)/g(~x). The expectation value of the result of this Monte Carlo integration, and its variance,