• Markov's chain
• Metropolis-Hasting
• non-gaussian background
• parameter estimation
• posterior
• unknown signal pdf

Gravitational waves are said to form a stochastic background when signals overlap and can’t be resolved as events. The overlap degree can be characterized by the duty cycle; this is defined as the ratio of the typical duration of the single event signal to the average distance between successive events. Astrophysical sources may form a stochastic background with duty cycle close to unity or less. This is a regime where maybe few events typically overlap and is called non-gaussian as opposed to the gaussian regime where many events overlap at each time point, hence the strain amplitude follows a normal distribution due to the central limit theorem. Among all sources, compact binary coalescences are expected to produce such a background based on last observations. The characterization of a non-gaussian background is reliant on the model and consists in estimating the model parameters. Parameters estimation can be performed generally using Bayesian methods and there are various tools for numerical estimates that can help the analysis, in this work we focalize on the Metropolis-Hasting algorithm. This is a powerful tool that uses Markov chains to generate samples from a non-necessarily normalized distribution. The problem that arises with models for non-gaussian backgrounds is that most times the probability distribution for the signal cannot be written in a closed mathematical form, while it is possible to computationally generate samples from it. Under the simplifying assumption that interferometers´ outputs are governed by gaussian noise, the probability distribution for the observations can be written as the expectation value of an observation dependent quantity over the probability distribution of the theoretical background. It is possible to approximate this expectation value using a montecarlo procedure, with a finite number of configurations, and then to use this approximate quantity to perform Bayesian parameter estimation. Specifically, a Random Walk Metropolis Hastings algorithm can be used to sample from posterior since classic numerical integration would be computationally probing, mostly in high parameters models. The fact that the algorithm runs using an estimator for the probability distribution, instead of its exact form, causes the sampling, and hence the produces posterior to have some bias from the real one. A toy model is used to try and characterize this method, it is a mixed gaussian model with two components, with three free parameters. The toy model is used to investigate computational efficiency of the method along with the bias question. Since the usual Metropolis algorithm presented some complications, i.e., the sampling getting stuck at some step because of the statistical fluctuations of the estimated posterior, a new slightly modified version, that solved this problem, was proposed with the name of “efficient sampling”. The difference consists in resampling at each step of the algorithm both the posterior of the current state and the posterior of the proposed state. In order to characterize the bias and its dependence on the number of realizations, many simulations were performed using different datasets produced with different number of detectors and different signal-to-noise ratios. An estimate of the convergence was done using Kullback–Leibler divergence. It was used in particular to compare an estimator for the posterior obtained histogramming the produced samples from metropolis, to the analytic, numerically-normalized posterior. The divergence dependence from the number of realizations was fitted with a complete power law. In all cases it was found that the fit converged to some value which is the same as the one obtained when calculating the divergence using the histogrammed samples generated from metropolis, but this time the analytic non-normalized posterior was used to make the sampling. This means that the residual bias can be attributed to other processes, but the convergence of the method is proven under the working assumptions, which are quite general. Also, the fitted parameters of the power law are showed to variate for different signal-to-noise ratio levels. The method is then applied to a large dataset of three coaligned, coincident interferometers with a low signal-to-noise ratio and averaging a relatively high number of times. Even under these more realistic conditions, the sampling performed well in reproducing the correct posterior. The high computational cost prohibited us from obtaining a large number of samples. Some possible reviews of the algorithm that could possibly increase the efficiency in both computational cost and obtained bias are studied in the last chapter. As an example, importance sampling can be used to more efficiently estimate the sampling distribution at each step. A future more complete development of the method could include the introduction of non-coaligned, non-coincident detectors.