The contributions of the Thesis can be summarized as follow:
1. Developments of the ABC–PMC algorithm to adaptively selecting the series of sequential tolerances 1:T = (1, 2, . . . , T) in order to improve the efficiency of the
sampling, along with an automatic stopping criterion that defines T (i.e. when to stop the algorithm). The proposed adaptive ABC–PMC tolerance selection algorithm can be easily implemented and examples are presented to show how this extension can improve not only the efficiency of the ABC–PMC algorithm but also avoiding to get stuck in local modes. This method, which works by evaluating the online performances of the ABC posterior distribution, is illustrated in Chapter 3. 2. Developments of the ABC–PMC algorithm as an alternative framework for in- ference to work with finite mixture models. There are several choices to take when implementing an ABC–PMC algorithm to work with finite mixture mod- els, including the selection of a suitable perturbation kernel to move the particles through the iterations (in particular to resample the mixture weights), how to address the label switching problem and the choice of high informative summary statistics. Beyond to discuss and address the required methodological extensions previously summarized, examples are presented to illustrate the performances of the proposed extended ABC–PMC algorithm to work with finite mixture models. This method is discussed in Chapter 4.
3. The CCF is an average of all the absorption lines of a stellar spectrum retrieved by using the radial velocity technique. Stellar activity can be probed by measuring variations in the shape of the CCF as function of time. Those variations are cal- culated using different parameters of the CCF. To measure with the best precision the necessary parameters is crucial to disentangle exoplanet signals from spurious variations in radial velocity caused by stellar activity. We propose to measure those parameters using a Skew Normal distribution, that compared to the Nor- mal distribution generally used, naturally includes an extra parameter to model the asymmetry of the CCF induced by convective blueshift. By using the Skew Normal distribution the barycenter and skewness of the CCF can be retrieved in a single operation, the correlation between the radial velocity of the star and stellar activity can be better understood and finally the uncertainties associated to all the parameters are smaller than the ones estimated with the classic analysis based on the Normal distribution. This method is presented in Chapter 5.
Chapter 2
Approximate Bayesian
Computation Methods
In this Chapter we introduce the ABC framework, the basic ABC algorithm and some of its already available extensions. Section 2.1 motivates the introduction for ABC as a statistical framework for inference. Starting from the basic ABC algorithm, in Section 2.2 we summarize the main challenges that need to be addressed in this framework in order to retrieve a suitable approximation of the true posterior distribution. In Section 2.3 one of the most famous extensions of the basic ABC algorithm, the ABC–PMC algorithm, is introduced. Our final remarks are outlined in Section 2.4.
2.1
Motivations for using Approximate Bayesian
Computation
Bayesian inference has become through the last two decades a suitable alternative to the frequentist approach. The relationship between the observed data yobs and parame-
ters θ ∈ Θ ⊆ Rp (i.e. p ≥ 1 is the dimension of the parameter space) can be described
by the likelihood function f (yobs | θ). In the Bayesian framework a prior distribution
has assigned to the vector of parameters θ ∼ π(θ), representing the subjective belief of the researcher. Bayesian inference is based on the resulting posterior distribution for θ, defined as: π(θ| yobs) = f (yobs| θ)π(θ) R Θf (yobs | θ)π(θ)dθ , (2.1)
where the denominator of Equation (2.1) is known also as the normalizing constant. If the elements of the posterior distribution in Equation (2.1) can be specified, then various techniques can be used to write down the posterior distribution exactly (e.g. if conjugate priors are specified) or approximated using various sampling techniques known
as Markov Chain Monte–Carlo (MCMC) algorithms, such as the Gibbs Sampling [59] and the Metropolis Hastings [65, 95].
Issues arise when the likelihood function cannot be specified. This happens for a variety of reasons such as the relationship between the data and the parameters is highly complex or unknown or if there are features of the data or data collecting procedure that are difficult to incorporate into a likelihood funtion (e.g. complex censuring or truncations). In the cases where it is not feasible to evaluate the likelihood function, ABC provides a framework for inference to obtain an approximation of the true posterior distribution.
In recent years ABC has been applied in many different fields of science, such as biology [137], ecology [7], epidemiology [92], population genetic problems [10, 33, 110, 135] and population modeling [140]. Given the complexity of its models and simulators, Astronomy seems to be a natural field for which an ABC based analysis could result really helpful in order to successfully address a variety of problems. Among the others, ABC has been used in problems such as the simulation of images for weak lensing measurements [1, 25], model analysis of morphological transformation of galaxies [26, 64], TYPE Ia supernovae [73, 75, 150] and for estimating of the luminosity function [128].