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The scope of this thesis is to propose new inferential tools, based on diffusion process approximations, for the study of the kinetic parameters in auto-regulatory networks. In the first part of this thesis, we study the applicability of the EA methodology to Stochastic Differential Equations (SDEs) which approximate biological systems. In principle EA can be applied to any scalar-valued SDE as long as a transformation (known as Lamperti transform) exists that sets the (new) infinitesimal variance to unity. We explore the numerical limitations of this requirement by considering a biological system that can be expressed as a scalar non-linear SDE. Next, we consider the multidimensional extension of this transformation and we show, with a counterexample, that EA can be applied to a class of SDEs which is wider than the class of reducible diffusions. In the second part of this thesis, we proposed a reparametrization of the kinetic constants that leads to an approximation known as the Linear Noise approximation (LNA). We prove that LNA converges to a linear SDE, as the size of the biological system increases. Since the LNA is a linear SDE, it has a known transition density with parameters given as the solutions of a system of Ordinary Differential Equations (ODEs) which are usually obtained numerically. Furthermore, we compare the LNA's simulation performance to the performance of other (approximate and exact) methods under different modelling scenarios and we relate the performance of the approximate methods to the system size. In addition, we consider LNA as an inferential tool and we use two methods, the Restarting (RE), which we propose, and the Non-Restarting (NR) method, proposed by Komorowski et. al. (2009) to derive the LNA's likelihood. The two methods differ on the initial conditions that they pose in order to solve the underlying ODEs. We compare the performance of the two methods by considering data generated under different scenarios. Finally, we discuss the lnar, a package for the R statistical environment, that we developed to implement the LNA methodology.

Markov Chain Monte Carlo (MCMC) methods often have difficulties in moving between isolated modes. To understand these difficulties, some MCMC theory and some mode jumping approaches will be reviewed, first in fixed dimension and later in variable dimension. The focus will lie on improving the eficiency of the powerful, but computationally expensive method "tempered transitions". A technique for optimising the method's parameters ("temperatures") will be proposed. It will be demonstrated that the default choice of geometric temperatures can be far from optimal. The tuning technique will then be tested on a hard applied sampling problem, namely on sampling from a fixed-dimensional mixture model. The results will show that the optimisation is robust and performs well and that tempered transitions achieves mode jumping ("label-switching") where standard MCMC fails. Since mixture models are often of variable dimension, it will be verified that tempered transitions and the tuning technique can also be applied in variable-dimensional problems. Tests on a variable-dimensional mixture model will confirm that tempered transitions also improves jumps between dimensions.

We consider a finite statespace continuous-time irreducible Markov chain (Χt)t≥0 together with some fluctuating additive functional (φt)t≥0. The objective is to condition the Markov process (Xt, φt)t≥0 on the event that the process (φt)t≥0 stays non-negative. There are three possible types of behaviour of the process (φt)t≥0: it can drift to +∞,oscillate, or drift to -∞, and in each of these cases we condition the process (Xt, φt)t≥0 on the event that the process (φt)t≥0 stays non-negative. In the positive drift case, the event that the process (φt)t≥0 stays non-negative is of positive probability and the process (Xt, φt)t≥0 can be conditioned on it in the standard way. In the oscillating and the negative drift cases, the event that the process (Xt, φt)t≥0 stays non-negative is of zero probability and we cannot condition the process (Xt, φt)t≥0 on it in the standard way. Instead, we look at the limits of laws of the process (Xt, φt)t≥0 conditioned on the event that the process (φt)t≥0 hits large levels before it crosses zero, and of laws of the process (Xt, φt)t≥0 conditioned on the event that the process (φt)t≥0 stays non-negative for a large time. In the oscillating case both limits exists and are equal to the same probability law. In the negative drift case, under certain conditions, both limits exist but give distinct probability laws. In addition, in the negative drift case, conditioning the process (Xt, φt)t≥0 on the event that the process (φt)t≥0 drifts to +∞ and then further conditioning on the event that the process (φt)t≥0 stays non-negative yields the same result as the limit of conditioning the process (Xt, φt)t≥0 on the event that the process (φt)t≥0 hits large levels before it crosses zero. Similarly, conditioning the process (Xt, φt)t≥0 on the event that the process (φt)t≥0 oscillates and then further conditioning on the event that the process (φt)t≥0 stays non-negative yields the same result as the limit of conditioning the process (Xt, φt)t≥0 on the event that the process (φt)t≥0 stays non-negative for a large time.

Avec la croissance exponentielle de la quantité de données, l’échantillonnage est une méthode pertinente pour étudier les populations. Parfois, nous avons besoin d’échantillonner un grand nombre d’objets d’une part pour exclure la possibilité d’un manque d’informations clés et d’autre part pour générer des résultats plus précis. Le problème réside dans le fait que l’échantillonnage d’un trop grand nombre d’individus peut constituer une perte de temps.Dans cette thèse, notre objectif est de chercher à établir des ponts entre la statistique et le k-processus ponctuel déterminantal(k-DPP) qui est défini via un noyau. Nous proposons trois projets complémentaires pour l’échantillonnage de grands ensembles de données en nous basant sur les k-DPPs. Le but est de sélectionner des ensembles variés qui couvrent un ensemble d’objets beaucoup plus grand en temps polynomial. Cela peut être réalisé en construisant différentes chaînes de Markov où les k-DPPs sont les lois stationnaires.Le premier projet consiste à appliquer les processus déterminantaux à la sélection d’espèces diverses dans un ensemble d’espèces décrites par un arbre phylogénétique. En définissant le noyau du k-DPP comme un noyau d’intersection, les résultats fournissent une borne polynomiale sur le temps de mélange qui dépend de la hauteur de l’arbre phylogénétique.Le second projet vise à utiliser le k-DPP dans un problème d’échantillonnage de sommets sur un graphe connecté de grande taille. La pseudo-inverse de la matrice Laplacienne normalisée est choisie d’étudier la vitesse de convergence de la chaîne de Markov créée pour l’échantillonnage de la loi stationnaire k-DPP. Le temps de mélange résultant est borné sous certaines conditions sur les valeurs propres de la matrice Laplacienne.Le troisième sujet porte sur l’utilisation des k-DPPs dans la planification d’expérience avec comme objets d’étude plus spécifiques les hypercubes latins d’ordre n et de dimension d. La clé est de trouver un noyau positif qui préserve le contrainte de ce plan c’est-à-dire qui préserve le fait que chaque point se trouve exactement une fois dans chaque hyperplan. Ensuite, en créant une nouvelle chaîne de Markov dont le n-DPP est sa loi stationnaire, nous déterminons le nombre d’étapes nécessaires pour construire un hypercube latin d’ordre n selon le n-DPP. / With the exponentially growing amount of data, sampling remains the most relevant method to learn about populations. Sometimes, larger sample size is needed to generate more precise results and to exclude the possibility of missing key information. The problem lies in the fact that sampling large number may be a principal reason of wasting time.In this thesis, our aim is to build bridges between applications of statistics and k-Determinantal Point Process(k-DPP) which is defined through a matrix kernel. We have proposed different applications for sampling large data sets basing on k-DPP, which is a conditional DPP that models only sets of cardinality k. The goal is to select diverse sets that cover a much greater set of objects in polynomial time. This can be achieved by constructing different Markov chains which have the k-DPPs as their stationary distribution.The first application consists in sampling a subset of species in a phylogenetic tree by avoiding redundancy. By defining the k-DPP via an intersection kernel, the results provide a fast mixing sampler for k-DPP, for which a polynomial bound on the mixing time is presented and depends on the height of the phylogenetic tree.The second application aims to clarify how k-DPPs offer a powerful approach to find a diverse subset of nodes in large connected graph which authorizes getting an outline of different types of information related to the ground set. A polynomial bound on the mixing time of the proposed Markov chain is given where the kernel used here is the Moore-Penrose pseudo-inverse of the normalized Laplacian matrix. The resulting mixing time is attained under certain conditions on the eigenvalues of the Laplacian matrix. The third one purposes to use the fixed cardinality DPP in experimental designs as a tool to study a Latin Hypercube Sampling(LHS) of order n. The key is to propose a DPP kernel that establishes the negative correlations between the selected points and preserve the constraint of the design which is strictly confirmed by the occurrence of each point exactly once in each hyperplane. Then by creating a new Markov chain which has n-DPP as its stationary distribution, we determine the number of steps required to build a LHS with accordance to n-DPP.

This research investigates existing relationships between the three apparently unrelated subjects: Markov process, Semigroups and Partial difierential equations. Markov processes define semigroups through their transition functions. Conversely particular semigroups determine transition functions and can be regarded as Markov processes. We have exploited these relationships to study some Markov chains. The infnitesimal generator of a Feller semigroup on the closure of a bounded domain of Rn; (n ^ 2), is an integro-diferential operator in the interior of the domain and verifes a boundary condition. The existence of a Feller semigroup defined by a diferential operator and a boundary condition is due to the existence of solution of a bounded value problem. From this result other existence suficient conditions on the existence of Feller semigroups have been obtained and we have applied some of them to construct Feller semigroups on the unity disk of R2. / Decision Sciences / M. Sc. (Operations Research)

This research investigates existing relationships between the three apparently unrelated subjects: Markov process, Semigroups and Partial difierential equations. Markov processes define semigroups through their transition functions. Conversely particular semigroups determine transition functions and can be regarded as Markov processes. We have exploited these relationships to study some Markov chains. The infnitesimal generator of a Feller semigroup on the closure of a bounded domain of Rn; (n ^ 2), is an integro-diferential operator in the interior of the domain and verifes a boundary condition. The existence of a Feller semigroup defined by a diferential operator and a boundary condition is due to the existence of solution of a bounded value problem. From this result other existence suficient conditions on the existence of Feller semigroups have been obtained and we have applied some of them to construct Feller semigroups on the unity disk of R2. / Decision Sciences / M. Sc. (Operations Research)