Nous étudions la régression bayésienne sous contraintes de régularité et de forme. Pour cela,on considère une base de B-spline pour obtenir une courbe lisse et nous démontrons que la forme d'une spline engendrée par une base de B-spline est contrôlée par un ensemble de points de contrôle qui ne sont pas situés sur la courbe de la spline. On propose différents types de contraintes de forme (monotonie, unimodalité, convexité, etc). Ces contraintes sont prises en compte grâce à la loi a priori. L'inférence bayésienne a permis de dériver la distribution posteriori sous forme explicite à une constante près. En utilisant un algorithme hybride de type Metropolis-Hastings avec une étape de Gibbs, on propose des simulations suivant la distribution a posteriori tronquée. Nous estimons la fonction de régression par le mode a posteriori. Un algorithme de type recuit simulé a permis de calculer le mode a posteriori. La convergence des algorithmes de simulations et du calcul de l'estimateur est prouvée. En particulier, quand les noeuds des B-splines sont variables, l'analyse bayésienne de la régression sous contrainte devient complexe. On propose des schémas de simulations originaux permettant de générer suivant la loi a posteriori lorsque la densité tronquée des coefficients de régression prend des dimensions variables. / We investigate the Bayesian regression under shape and smoothness constraints. We first elicita Bayesian method for regression under shape restrictions and smoothness conditions. Theregression function is built from B-spline basis that controls its regularity. Then we show thatits shape can be controlled simply from its coefficients in the B-spline basis. This is achievedthrough the control polygon whose definition and some properties are given in this article.The regression function is estimated by the posterior mode. This mode is calculated by asimulated annealing algorithm which allows to take into account the constraints of form inthe proposal distribution. A credible interval is obtained from simulations using Metropolis-Hastings algorithm with the same proposal distribution as the simulated annealing algorithm.The convergence of algorithms for simulations and calculation of the estimator is proved. Inparticular, in the case of Bayesian regression under constraints and with free knots, Bayesiananalysis becomes complex. we propose original simulation schemes which allows to simulatefrom the truncated posterior distribution with free dimension.

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The premise of this report is to delve into sports betting and whether favourable opportunities can be found, more specifically focusing on over and under odds for number of goals scored in football games of the Premier League. Using historical data from football matches several models are developed, the characteristics of goals warranting the use of probability based Poisson and Negative Binomial models, as well as Bayesian Poisson regression for goal predictions. Once these models were developed odds was found and compared to bookmakers, the results indicated that all models, to varying degrees, find favourable opportunities and profitable betting strategies can be identified. This suggests that bookmakers do not always price betting products according to their true probabilities likely due to book balancing and informational asymmetries. Furthermore it indicates that there is a presence of inefficiencies in the sports betting market. / Den här rapporten kommer djupdyka i betting och huruvida gynnsamma möjligheter kan hittas. Mer specifikt kommer ett fokus ligga på över/under odds för antalet mål i fotbollsmatcher i engelska Premier League. Genom att använda historisk data från fotbollsmatcher utvecklas flera olika statistiska modeller för att förutspå antalet mål i fotbollsmatcher. Skattning av Poisson och Negativ Binomial fördelningar samt utvecklandet av en Bayesiansk Poisson regressionsmodell motiveras av egenskaperna hos antalet mål i fotbollsmatcher. Med dessa modeller, beräknas odds för flera framtida matcher inom Premier League och dessa jämfördes med odds som ges av bettingbolag. Resultaten indikerar att alla modeller kan, i olika stor utsträckning, hitta gynnsamma möjligheter och lönsamma betting strategier kan identifieras. Detta tyder på att bettingbolag inte alltid sätter sina odds enbart baserat på den faktiska sannolikheten, vilket troligtvis beror på att bolagen balanserar sina böcker samt informationsasymmetrier. Dessutom indikerar resultatet på att det finns faktorer på bettingmarknaden som gör marknaden ineffektiv.