Using the Maximum Likelihood Method to find the Optimal Parameters of the Models Describing the Pulsar Population Statistics
Dr. Peter Gonthier, Hope College
We seek to maximize the likelihood of a set of parameters describing the distributions of the initial period, magnetic field, and radio luminosities in our pulsar population simulation code (Harding, Grenier, and Gonthier 2007) and in order to better understand the confidence region of the model. Our pulsar population code simulates pulsars at birth using Monte Carlo techniques and evolves them to the present assuming different models describing the birth distributions, spin down, and emission. One problem with this method is that it is difficult to explore the parameter space due to high computational time. Since we are dealing with comparisons of low counts, we need to describe the distributions using Poisson statistics, which is best accomplished within the maximum likelihood method (MLM). The popular least-squares method is an instance of the MLM that assumes Gaussian statistics, a valid assumption when the number of counts is high. In this scenario, we seek to maximize the likelihood using a Monte Carlo Markov Chain algorithm that randomly explores the parameter space. An efficient method to accomplish this is the Hybrid Monte Carlo algorithm (HMC), a means of searching the parameter space by making jumps towards areas with higher likelihood. The observed pulsar characteristics that we will be trying to fit consist of the period, period derivative, the dispersion measure, and the radio flux. However, before the application to the main goal associated with the population study, a simple case study is presented as an illustration of the method.
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