Bayesian inference for inverse problems

Abstract

Abstract. Traditionally, the MaxEnt workshops start by a tutorial day. This paper summarizes my talk during 2001'th workshop at John Hopkins University. The main idea in this talk is to show how the Bayesian inference can naturally give us all the necessary tools we need to solve real inverse problems: starting by simple inversion where we assume to know exactly the forward model and all the input model parameters up to more realistic advanced problems of myopic or blind inversion where we may be uncertain about the forward model and we may have noisy data.Starting by an introduction to inverse problems through a few examples and explaining their ill posedness nature, I briefly presented the main classical deterministic methods such as data matching and classical regularization methods to show their limitations. I then presented the main classical probabilistic methods based on likelihood, information theory and maximum entropy and the Bayesian inference framework for such problems. I show that the Bayesian framework, not only generalizes all these methods, but also gives us natural tools, for example, for inferring the uncertainty of the computed solutions, for the estimation of the hyperparameters or for handling myopic or blind inversion problems. Finally, through a deconvolution problem example, I presented a few state of the art methods based on Bayesian inference particularly designed for some of the mass spectrometry data processing problems.

Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems

“…The resulting joint posterior over model parameters and hyperparameters may then be interrogated in various ways-e.g., by marginalizing over the hyperparameters to obtain pðmjdÞ; or first marginalizing over m and using the maximizer of this density as an estimate of the hyperparameters; or by seeking the joint maximum a posteriori estimate or posterior mean of m, / m , and / g [54,48]. In the present study, we will introduce hyperparameters to describe aspects of the prior covariance.…”

“…Bayesian approaches to inverse problems have received much recent interest [48,49,4], with applications ranging from geophysics [50,51] and climate modeling [52] to heat transfer [53,20]. We review this approach briefly below; for more extensive introductions, see [4,5,48].…”

“…If parameters / m of the prior density p m ðmj/ m Þ or parameters / g of the error model p g ðg i j/ g Þ are not known a priori, they may become additional objects for Bayesian inference. In other words, these hyperparameters may themselves be endowed with priors and estimated from data [48]: pðm; / m ; / g jdÞ / pðdjm; / g Þp m ðmj/ m Þpð/ g Þpð/ m Þ: ð20Þ…”

Stochastic spectral methods for efficient Bayesian solution of inverse problems

“…The Bayesian setting for inverse problems offers a rigorous foundation for inference from noisy data and uncertain forward models, a natural mechanism for incorporating prior information, and a quantitative assessment of uncertainty in the inferred results [3,4]. Indeed, the output of Bayesian inference is not a single value for the model parameters, but a probability distribution that summarizes all available information about the parameters.…”

Bridging the gap between atomistic phenomena and continuum behavior in electrochemical energy storage processes.

“…The variance σ 2 of the discrepancy is related to the fluctuation in the flux which level depends on the temperature. σ 2 is expected to decrease inversely proportional to the number of samples N and will be inferred as a hyperparameter [65,62] along with A and B. Bayes' rule is then written as:…”