Bayesian Analysis of Linear Inverse Problems with Applications in Economics and Finance

In my PhD thesis I propose a Bayesian nonparametric estimation method for structural econometric models where the functional parameter of interest describes the economic agent's behavior. The structural parameter is characterized as the solution of a functional equation, or by using more technical words, as the solution of an inverse problem that can be either ill-posed or well-posed. From a Bayesian point of view, the parameter of interest is a random function and the solution to the inference problem is the posterior distribution of this parameter. A regular version of the posterior distribution in functional spaces is characterized. However, the infinite dimension of the considered spaces causes a problem of non continuity of the solution and then a problem of inconsistency, from a frequentist point of view, of the posterior distribution (i.e. problem of ill-posedness). The contribution of this essay is to propose new methods to deal with this problem of ill-posedness. The first one consists in adopting a Tikhonov regularization scheme in the construction of the posterior distribution so that I end up with a new object that I call regularized posterior distribution and that I guess it is solution of the inverse problem. The second approach consists in specifying a prior distribution on the parameter of interest of the g-prior type. Then, I detect a class of models for which the prior distribution is able to correct for the ill-posedness also in infinite dimensional problems. I study asymptotic properties of these proposed solutions and I prove that, under some regularity condition satisfied by the true value of the parameter of interest, they are consistent in a "frequentist" sense. Once I have set the general theory, I apply my bayesian nonparametric methodology to different estimation problems. First, I apply this estimator to deconvolution and to hazard rate, density and regression estimation. Then, I consider the estimation of an Instrumental Regression that is useful in micro-econometrics when we have to deal with problems of endogeneity. Finally, I develop an application in finance: I get the bayesian estimator for the equilibrium asset pricing functional by using the Euler equation defined in the Lucas'(1978) tree-type models.

Supramolecular functional assemblies with photo or chemical addressability

Supramolecular self-assembly represents a key technology for the spontaneous construction of nanoarchitectures and for the fabrication of materials with enhanced physical and chemical properties. In addition, a significant asset of supramolecular self-assemblies rests on their reversible formation, thanks to the kinetic lability of their non-covalent interactions. This dynamic nature can be exploited for the development of “self-healing” and “smart” materials towards the tuning of their functional properties upon various external factors. One particular intriguing objective in the field is to reach a high level of control over the shape and size of the supramolecular architectures, in order to produce well-defined functional nanostructures by rational design. In this direction, many investigations have been pursued toward the construction of self-assembled objects from numerous low-molecular weight scaffolds, for instance by exploiting multiple directional hydrogen-bonding interactions. In particular, nucleobases have been used as supramolecular synthons as a result of their efficiency to code for non-covalent interaction motifs. Among nucleobases, guanine represents the most versatile one, because of its different H-bond donor and acceptor sites which display self-complementary patterns of interactions. Interestingly, and depending on the environmental conditions, guanosine derivatives can form various types of structures. Most of the supramolecular architectures reported in this Thesis from guanosine derivatives require the presence of a cation which stabilizes, via dipole-ion interactions, the macrocyclic G-quartet that can, in turn, stack in columnar G-quadruplex arrangements. In addition, in absence of cations, guanosine can polymerize via hydrogen bonding to give a variety of supramolecular networks including linear ribbons. This complex supramolecular behavior confers to the guanine-guanine interactions their upper interest among all the homonucleobases studied. They have been subjected to intense investigations in various areas ranging from structural biology and medicinal chemistry – guanine-rich sequences are abundant in telomeric ends of chromosomes and promoter regions of DNA, and are capable of forming G-quartet based structures– to material science and nanotechnology. This Thesis, organized into five Chapters, describes mainly some recent advances in the form and function provided by self-assembly of guanine based systems. More generally, Chapter 4 will focus on the construction of supramolecular self-assemblies whose self-assembling process and self-assembled architectures can be controlled by light as external stimulus. Chapter 1 will describe some of the many recent studies of G-quartets in the general area of nanoscience. Natural G- quadruplexes can be useful motifs to build new structures and biomaterials such as self-assembled nanomachines, biosensors, therapeutic aptamer and catalysts. In Chapters 2-4 it is pointed out the core concept held in this PhD Thesis, i.e. the supramolecular organization of lipophilic guanosine derivatives with photo or chemical addressability. Chapter 2 will mainly focus on the use of cation-templated guanosine derivatives as a potential scaffold for designing functional materials with tailored physical properties, showing a new way to control the bottom-up realization of well-defined nanoarchitectures. In section 2.6.7, the self-assembly properties of compound 28a may be considered an example of open-shell moieties ordered by a supramolecular guanosine architecture showing a new (magnetic) property. Chapter 3 will report on ribbon-like structures, supramolecular architectures formed by guanosine derivatives that may be of interest for the fabrication of molecular nanowires within the framework of future molecular electronic applications. In section 3.4 we investigate the supramolecular polymerizations of derivatives dG 1 and G 30 by light scattering technique and TEM experiments. The obtained data reveal the presence of several levels of organization due to the hierarchical self-assembly of the guanosine units in ribbons that in turn aggregate in fibrillar or lamellar soft structures. The elucidation of these structures furnishes an explanation to the physical behaviour of guanosine units which display organogelator properties. Chapter 4 will describe photoresponsive self-assembling systems. Numerous research examples have demonstrated that the use of photochromic molecules in supramolecular self-assemblies is the most reasonable method to noninvasively manipulate their degree of aggregation and supramolecular architectures. In section 4.4 we report on the photocontrolled self-assembly of modified guanosine nucleobase E-42: by the introduction of a photoactive moiety at C8 it is possible to operate a photocontrol over the self-assembly of the molecule, where the existence of G-quartets can be alternately switched on and off. In section 4.5 we focus on the use of cyclodextrins as photoresponsive host-guest assemblies: αCD–azobenzene conjugates 47-48 (section 4.5.3) are synthesized in order to obtain a photoresponsive system exhibiting a fine photocontrollable degree of aggregation and self-assembled architecture. Finally, Chapter 5 contains the experimental protocols used for the research described in Chapters 2-4.

Lesion detection and classification in breast cancer: evaluation of approaches based on morphological features, tracer kinetic modelling and semi-quantitative parameters in MR functional imaging (DCE-MRI)

The diagnosis, grading and classification of tumours has benefited considerably from the development of DCE-MRI which is now essential to the adequate clinical management of many tumour types due to its capability in detecting active angiogenesis. Several strategies have been proposed for DCE-MRI evaluation. Visual inspection of contrast agent concentration curves vs time is a very simple yet operator dependent procedure, therefore more objective approaches have been developed in order to facilitate comparison between studies. In so called model free approaches, descriptive or heuristic information extracted from time series raw data have been used for tissue classification. The main issue concerning these schemes is that they have not a direct interpretation in terms of physiological properties of the tissues. On the other hand, model based investigations typically involve compartmental tracer kinetic modelling and pixel-by-pixel estimation of kinetic parameters via non-linear regression applied on region of interests opportunely selected by the physician. This approach has the advantage to provide parameters directly related to the pathophysiological properties of the tissue such as vessel permeability, local regional blood flow, extraction fraction, concentration gradient between plasma and extravascular-extracellular space. Anyway, nonlinear modelling is computational demanding and the accuracy of the estimates can be affected by the signal-to-noise ratio and by the initial solutions. The principal aim of this thesis is investigate the use of semi-quantitative and quantitative parameters for segmentation and classification of breast lesion. The objectives can be subdivided as follow: describe the principal techniques to evaluate time intensity curve in DCE-MRI with focus on kinetic model proposed in literature; to evaluate the influence in parametrization choice for a classic bi-compartmental kinetic models; to evaluate the performance of a method for simultaneous tracer kinetic modelling and pixel classification; to evaluate performance of machine learning techniques training for segmentation and classification of breast lesion.