### abstract

- Bayesian inference is based on three evidence components: experimental observations, model predictions and expert's beliefs. Integrating experimental evidence into the calibration or selection of a model, either empirically of physically based, is of great significance in almost every area of science and engineering because it maps the response of the process of interest into a set of parameters, which aim at explaining the process' governing characteristics. This work introduces the use of the Bayesian paradigm to construct full probabilistic description of parameters of spatial processes. The influence of uncertainty is first discussed on the calibration of an empirical relationship between remolded undrained shear strength Su-r and liquidity index IL, as a potential predictor of the soil strength. Two site-specific datasets are considered in the analysis. The key emphasis of the study is to construct a unified regression model reflecting the characteristics of the both contributing data sets, while the site dependency of the data is properly accounted for. We question the regular Bayesian updating process, since a test of statistics proves that the two data sets belong to different populations. Application of "Disjunction" probability operator is proposed as an alternative to arrive at a more conclusive Su-r-IL model. Next, the study is extended to a functional inverse problem where the object of inference constructs a spatial random field. We introduce a methodology to infer the spatial variation of the elastic characteristics of a heterogeneous earth model via Bayesian approach, given the probed medium's response to interrogating waves measured at the surface. A reduced dimension, self regularized treatment of the inverse problem using partition modeling is introduced, where the velocity field is discretized by a variable number of disjoint regions defined by Voronoi tessellations. The number of partitions, their geometry and weights dynamically vary during the inversion, in order to recover the subsurface image. The idea of treating the number of tessellation (number of parameters) as a parameter itself is closely associated with probabilistic model selection. A reversible jump Markov chain Monte Carlo (RJMCMC) scheme is applied to sample the posterior distribution of varying dimension. Lastly, direct treatment of a Bayesian model selection through the definition of the Bayes factor (BF) is developed for linear models, where it is employed to define the most likely order of the virial Equation of State (EOS). Virial equation of state is a constitutive model describing the thermodynamic behavior of low-density fluids in terms of the molar density, pressure and temperature. Bayesian model selection has successfully determined the best EOS that describes four sets of isotherms, where approximate (BIC) method either failed to select a model or fevered an overly-flexible model, which specifically perform poorly in terms of prediction.