Henning Omre, profesor de la Norwegian University of Science et Technology, Noruega

Impartirá la conferencia Bayesian Sptatial Inversion with conjugate selection gaussian prior models.

Día: 22 de noviembre de 2017,

Hora: 12:00

Lloc: Saló de Graus Facultat de Matemàtiques.

Bayesian Spatial Inversion With Conjugate Selection Gaussian Prior Models

Abstract

Bayesian inversion is defined in a predictive setting. Observations are acquired from 
a spatial variable of interest, often through a complex acquisition procedure with random
components. This procedure defines the observation likelihood model. The objective is to 
assess the spatial variable from the set of observations. A prior probabilistic model
must be defined for the spatial variable of interest and this model should capture 
known characteristics about the variable. The posterior model, being the solution to 
Bayesian inversion, is uniquely defined by the likelihood and prior models.  For 
high-dimensional spatial variables the normalizing constant in the posterior model 
can usually not be calculated. Moreover, assessment of the posterior model by
brute-force Markov chain Monte Carlo simulation is normally prohibited due to 
high dimensions and strong coupling in the model.

In order to facilitate assessment of the posterior model the consept of conjugate
prior models in Bayesian spatial inversion is introduced. In classical Bayesian inference, 
focusing on model parameter estimation, this consept is familiar. For a given 
likelihood model, a prior model from the associate class of conjugate pdfs ensures
that the corresponding posterior model belongs to the same class of pdfs. The 
challenge is larger in Bayesian spatial inversion, however, since the variety of 
likelihood models are much larger and the prior models must be high-dimensional 
random fields and not merely a low-dimensional vector of model parameters.

Spatial variables represented by continuous random fields are 
considered. For a likelihood model being Gauss-linear the conjugate class is
Gaussian random fields. The advantage of using conjugate prior 
models is that efficient procedures for assessing the corresponding posterior 
models and for model parameter inference can be defined.

The major contribution of the presentation is the definition an selection operator
that can be activated on conjugate prior models, for which the resulting prior model 
will also be a conjugate prior model.  Hence, if the prior Gaussian
random field which is a conjugate class with respect to Gauss-linear likelihood models
is subject to this selection operator also the resulting selection Gaussian random 
field will be a conjugate class with respect to the same likelihood model. This closedness 
property is demonstrated both for assessing the posterior model and for model parameter 
inference.

Examples from seismic inversion of real seismic data from the North Sea will be presented.