Recursion is a fundamental tool of mathematics used to define, construct, and analyze mathematical objects. This work employs induction, sieving, inversion, and other recursive methods to solve a variety of problems in the areas of algebraic number theory, topological and combinatorial graph theory, and analytic probability and statistics. A common theme of recursively defined functions, weighted sums, and cross-referencing sequences arises in all three contexts, and supplemented by sieving methods, generating functions, asymptotics, and heuristic algorithms. In the area of number theory, this work generalizes the sieve of Eratosthenes to a sequence of polynomial values called polynomial-value sieving. In the case of quadratics, the method of polynomial-value sieving may be characterized briefly as a product presentation of two binary quadratic forms. Polynomials for which the polynomial-value sieving yields all possible integer factorizations of the polynomial values are called recursively-factorable. The Euler and Legendre prime producing polynomials of the form n2+n+p and 2n2+p, respectively, and Landau's n2+1 are shown to be recursively-factorable. Integer factorizations realized by the polynomial-value sieving method, applied to quadratic functions, are in direct correspondence with the lattice point solutions (X,Y) of the conic sections aX2+bXY +cY2+X-nY=0. The factorization structure of the underlying quadratic polynomial is shown to have geometric properties in the space of the associated lattice point solutions of these conic sections. In the area of combinatorial graph theory, this work considers two topological structures that are used to model the process of homologous genetic recombination: assembly graphs and chord diagrams. The result of a homologous recombination can be recorded as a sequence of signed permutations called a micronuclear arrangement. In the assembly graph model, each micronuclear arrangement corresponds to a directed Hamiltonian polygonal path within a directed assembly graph. Starting from a given assembly graph, we construct all the associated micronuclear arrangements. Another way of modeling genetic rearrangement is to represent precursor and product genes as a sequence of blocks which form arcs of a circle. Associating matching blocks in the precursor and product gene with chords produces a chord diagram. The braid index of a chord diagram can be used to measure the scope of interaction between the crossings of the chords. We augment the brute force algorithm for computing the braid index to utilize a divide and conquer strategy. Both assembly graphs and chord diagrams are closely associated with double occurrence words, so we classify and enumerate the double occurrence words based on several notions of irreducibility. In the area of analytic probability, moments abstractly describe the shape of a probability distribution. Over the years, numerous varieties of moments such as central moments, factorial moments, and cumulants have been developed to assist in statistical analysis. We use inversion formulas to compute high order moments of various types for common probability distributions, and show how the successive ratios of moments can be used for distribution and parameter fitting. We consider examples for both simulated binomial data and the probability distribution affiliated with the braid index counting sequence. Finally we consider a sequence of multiparameter binomial sums which shares similar properties with the moment sequences generated by the binomial and beta-binomial distributions. This sequence of sums behaves asymptotically like the high order moments of the beta distribution, and has completely monotonic properties.

Nous établissons des conditions nécessaires et suffisantes d’existence et d’infinie divisibilité pour des processus ponctuels alpha-déterminantaux et, lorsque alpha est positif, pour leur intensité sous-jacente (en tant que processus de Cox). Dans le cas où l’espace est fini, ces distributions correspondent à des lois binomiales, négatives binomiales et gamma multidimensionnelles. Nous étudions de façon approfondie ces deux derniers cas avec un noyau non nécessairement symétrique. / We establish necessary and sufficient conditions for the existence and infinite divisibility of alpha-determinantal processes and, when alpha is positive, of their underlying intensity (as Cox process). When the space is finite, these distributions correspond to multidimensional binomial, negative binomial and gamma distributions. We make an in-depth study of these last two cases with a non necessarily symmetric kernel.

<p>Graphs are a natural choice for understanding the associations between variables, and assuming a probabilistic embedding for the graph structure leads to a variety of graphical models that enable us to understand these associations even further. In the realm of high-dimensional data, where the number of associations between interacting variables is far greater than the available number of data points, the goal is to infer a sparse graph. In this thesis, we make contributions in the domain of Bayesian graphical models, where our prior belief on the graph structure, encoded via uncertainty on the model parameters, enables the estimation of sparse graphs.</p> <p><br></p> <p>We begin with the Gaussian Graphical Model (GGM) in Chapter 2, one of the simplest and most famous graphical models, where the joint distribution of interacting variables is assumed to be Gaussian. In GGMs, the conditional independence among variables is encoded in the inverse of the covariance matrix, also known as the precision matrix. Under a Bayesian framework, we propose a novel prior--penalty dual called the `graphical horseshoe-like' prior and penalty, to estimate precision matrix. We also establish the posterior convergence of the precision matrix estimate and the frequentist consistency of the maximum a posteriori (MAP) estimator.</p> <p><br></p> <p>In Chapter 3, we develop a general framework based on local linear approximation for MAP estimation of the precision matrix in GGMs. This general framework holds true for any graphical prior, where the element-wise priors can be written as a Laplace scale mixture. As an application of the framework, we perform MAP estimation of the precision matrix under the graphical horseshoe penalty.</p> <p><br></p> <p>In Chapter 4, we focus on graphical models where the joint distribution of interacting variables cannot be assumed Gaussian. Motivated by the quantile graphical models, where the Gaussian likelihood assumption is relaxed, we draw inspiration from the domain of precision medicine, where personalized inference is crucial to tailor individual-specific treatment plans. With an aim to infer Directed Acyclic Graphs (DAGs), we propose a novel quantile DAG learning framework, where the DAGs depend on individual-specific covariates, making personalized inference possible. We demonstrate the potential of this framework in the regime of precision medicine by applying it to infer protein-protein interaction networks in Lung adenocarcinoma and Lung squamous cell carcinoma.</p> <p><br></p> <p>Finally, we conclude this thesis in Chapter 5, by developing a novel framework to compute the marginal likelihood in a GGM, addressing a longstanding open problem. Under this framework, we can compute the marginal likelihood for a broad class of priors on the precision matrix, where the element-wise priors on the diagonal entries can be written as gamma or scale mixtures of gamma random variables and those on the off-diagonal terms can be represented as normal or scale mixtures of normal. This result paves new roads for model selection using Bayes factors and tuning of prior hyper-parameters.</p>

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