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Concept# Maximum a posteriori estimation

Summary

In Bayesian statistics, a maximum a posteriori probability (MAP) estimate is an estimate of an unknown quantity, that equals the mode of the posterior distribution. The MAP can be used to obtain a point estimate of an unobserved quantity on the basis of empirical data. It is closely related to the method of maximum likelihood (ML) estimation, but employs an augmented optimization objective which incorporates a prior distribution (that quantifies the additional information available through prior knowledge of a related event) over the quantity one wants to estimate. MAP estimation can therefore be seen as a regularization of maximum likelihood estimation.
Description
Assume that we want to estimate an unobserved population parameter \theta on the basis of observations x. Let f be the sampling distribution of x, so that f(x\mid\theta) is the probability of x when the underlying population parameter

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This thesis is devoted to the derivation of error estimates for partial differential equations with random input data, with a focus on a posteriori error estimates which are the basis for adaptive strategies. Such procedures aim at obtaining an approximation of the solution with a given precision while minimizing the computational costs. If several sources of error come into play, it is then necessary to balance them to avoid unnecessary work. We are first interested in problems that contain small uncertainties approximated by finite elements. The use of perturbation techniques is appropriate in this setting since only few terms in the power series expansion of the exact random solution with respect to a parameter characterizing the amount of randomness in the problem are required to obtain an accurate approximation. The goal is then to perform an error analysis for the finite element approximation of the expansion up to a certain order. First, an elliptic model problem with random diffusion coefficient with affine dependence on a vector of independent random variables is studied. We give both a priori and a posteriori error estimates for the first term in the expansion for various norms of the error. The results are then extended to higher order approximations and to other sources of uncertainty, such as boundary conditions or forcing term. Next, the analysis of nonlinear problems in random domains is proposed, considering the one-dimensional viscous Burgers' equation and the more involved incompressible steady-state Navier-Stokes equations. The domain mapping method is used to transform the equations in random domains into equations in a fixed reference domain with random coefficients. We give conditions on the mapping and the input data under which we can prove the well-posedness of the problems and give a posteriori error estimates for the finite element approximation of the first term in the expansion. Finally, we consider the heat equation with random Robin boundary conditions. For this parabolic problem, the time discretization brings an additional source of error that is accounted for in the error analysis. The second part of this work consists in the analysis of a random elliptic diffusion problem that is approximated in the physical space by the finite element method and in the stochastic space by the stochastic collocation method on a sparse grid. Considering a random diffusion coefficient with affine dependence on a vector of independent random variables, we derive a residual-based a posteriori error estimate that controls the two sources of error. The stochastic error estimator is then used to drive an adaptive sparse grid algorithm which aims at alleviating the so-called curse of dimensionality inherent to tensor grids. Several numerical examples are given to illustrate the performance of the adaptive procedure.

Powerful mathematical tools have been developed for trading in stocks and bonds, but other markets that are equally important for the globalized world have to some extent been neglected. We decided to study the shipping market as an new area of development in mathematical finance. The market in shipping derivatives (FFA and FOSVA) has only been developed after 2000 and now exhibits impressive growth. Financial actors have entered the field, but it is still largely undiscovered by institutional investors. The first part of the work was to identify the characteristics of the market in shipping, i.e. the segmentation and the volatility. Because the shipping business is old-fashioned, even the leading actors on the world stage (ship owners and banks) are using macro-economic models to forecast the rates. If the macro-economic models are logical and make sense, they fail to predict. For example, the factor port congestion has been much cited during the last few years, but it is clearly very difficult to control and is simply an indicator of traffic. From our own experience it appears that most ship owners are in fact market driven and rather bad at anticipating trends. Due to their ability to capture large moves, we chose to consider Lévy processes for the underlying price process. Compared with the macro-economic approach, the main advantage is the uniform and systematic structure this imposed on the models. We get in each case a favorable result for our technology and a gain in forecasting accuracy of around 10% depending on the maturity. The global distribution is more effectively modelled and the tails of the distribution are particularly well represented. This model can be used to forecast the market but also to evaluate the risk, for example, by computing the VaR. An important limitation is the non-robustness in the estimation of the Lévy processes. The use of robust estimators reinforces the information obtained from the observed data. Because maximum likelihood estimation is not easy to compute with complex processes, we only consider some very general robust score functions to manage the technical problems. Two new class of robust estimators are suggested. These are based on the work of F. Hampel ([29]) and P. Huber ([30]) using influence functions. The main idea is to bound the maximum likelihood score function. By doing this a bias is created in the parameters estimation, which can be corrected by using a modification of the following type and as proposed by F. Hampel. The procedure for finding a robust estimating equation is thus decomposed into two consecutive steps : Subtract the bias correction and then Bound the score function. In the case of complex Lévy processes, the bias correction is difficult to compute and generally unknown. We have developed a pragmatic solution by inverting the Hampel's procedure. Bound the score function and then Correct for the bias. The price is a loss of the theoretical properties of our estimators, besides the procedure converges to maximum likelihood estimate. A second solution to for achieving robust estimation is presented. It considers the limiting case when the upper and lower bounds tend to zero and leads to B-robust estimators. Because of the complexity of the Lévy distributions, this leads to identification problems.

Multiple generalized additive models are a class of statistical regression models wherein parameters of probability distributions incorporate information through additive smooth functions of predictors. The functions are represented by basis function expansions, whose coefficients are the regression parameters. The smoothness is induced by a quadratic roughness penalty on the functionsâ curvature, which is equivalent to a weighted $L_2$ regularization controlled by smoothing parameters. Regression fitting relies on maximum penalized likelihood estimation for the regression coefficients, and smoothness selection relies on maximum marginal likelihood estimation for the smoothing parameters.
Owing to their nonlinearity, flexibility and interpretability, generalized additive models are widely used in statistical modeling, but despite recent advances, reliable and fast methods for automatic smoothing in massive datasets are unavailable. Existing approaches are either reliable, complex and slow, or unreliable, simpler and fast, so a compromise must be made. A bridge between these categories is needed to extend use of multiple generalized additive models to settings beyond those possible in existing software. This thesis is one step in this direction. We adopt the marginal likelihood approach to develop approximate expectation-maximization methods for automatic smoothing, which avoid evaluation of expensive and unstable terms. This results in simpler algorithms that do not sacrifice reliability and achieve state-of-the-art accuracy and computational efficiency.
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