# How are Bayesian deconvolution, Bayesian inverse problem, Inverse Bayesian, Prior elicitation different?

This question started from discussion in naming of SBC package on how to name family of optimizers such as HMC, VI: estimator or computational algorithm.

I am confused by similar concepts with different names in the subject line. To ask for feedback, I summarized my search so far, longing to boost other searching souls including my future self :)

Bayesian deconvolution is introduced in

Bayesian inverse probelm

Focus is more on posterior inference in inverse probelm. I read this on Bayesian inversion, Sec.4 Prior construction from this and Chp.21 Empirical Bayes Estimation Strategies from Efron’s textbook.

Inverse Bayesian
For rare events with few data, first assume data distribution as posterior and infer marginal prior (Wagner et al., 2021; Wahal and Biros., 2019; Rao et al., 2020). Posterior cannot be expressed with canonical distribution (e.g. Normal, Gamma, etc.), hence using Kalman filter and Gaussian process are their novelty.

Prior elicitation
Stan community can provide deepest level of prior knowledge for me on this word e.g. @avehtari @paul.buerkner and other’s paper on prior knowledge elicitation. Compared to the above concepts which is concentrated on posterior, focus of this word seems to be on prior. However, from Bayesian workflow lens, whose point is iterating till accepted (assumption sets with test quantities), its goal doesn’t seem to have much difference with the others.

If anyone could kindly share knowledge on their differences, history, usage it would be helpful. Thanks.

Historically “deconvolution” and “inverse problems” (and related terms like “uncertainty quantification”) have referred to particular applications and the models typically used for those applications. More formally consider a model were data scatter around the output of some complex, many-to-one function,

y \sim \text{normal}(f(x), \sigma).

Here x is the unobserved behavior of interest and f is the “forward model” that quantifies which features manifest in the output . For example x might be a latent image with f quantifying the various ways in which the image is warped and obscured by the camera or x might be an initial state with f a dynamical system that chaotically evolves the initial state to a final state.

Because f is many-to-one naively inverting the forward model to observed data is ill-posed: f^{-1}(\tilde{y}) does not result in a single value for x but rather range of values. The language isn’t consistent but I would say that “deconvolution” is used more to refer to regularizing f^{-1}(\tilde{y}) to a single point estimate of x while “inverse problems” refers to quantifying the full geometry of f^{-1}(\tilde{y}).

Bayesian deconvolution and inverse problems then typically refers to Bayesian inference over these ill-posed systems, sometimes focusing on quantifying how f^{-1}(\tilde{y}) manifests in the posterior distribution and sometimes focusing on regularizing f^{-1}(\tilde{y}) with informative prior models so that the posterior distribution is less degenerate than f^{-1}(\tilde{y}). When that informative prior model is motivated by f^{-1}(\tilde{y}) itself, and not actual domain expertise, this becomes a form of empirical Bayes.

I looked only briefly but Wagner et al., 2021 and Wahal and Biros., 2019 both seem to refer to inverse problems of this sort. Wahal and Biros complicate things by also referring to their particular computational method (which focuses on quantifying posterior tail behavior) as “inverse Monte Carlo”.

Historically “prior elicitation” is a bit more precise. It refers to the various ways that one can extract information from oneself, collaborators, colleagues, literature, and the like in order to inform a prior model.

In the particular context of inverse problems one could focus prior elicitation on information that would help to regularize the ill-posed inverse regularizing f^{-1}(\tilde{y}). In other words focusing elicitation on informing those parameters that are poorly informed by f^{-1}(\tilde{y}) while not worrying about those that are informed. That said this kind of conditional prior elicitation has be performed very carefully. The structure of f^{-1}(\tilde{y}) from any particular observation can help motivate what kind of information we should elicit (i.e. about which parameters we need more information). If it is used to make that information up then it becomes a form of empirical Bayes which used the data twice to construct a posterior distribution.

Again the language is all over the place but I would say that:

Deconvolution, inverse problem, and uncertainty quantification are most universally used to refer to a many-to-one forward model that results in ill-posed inferences. They are sometimes also used to refer to particular methods for regularizing those inferences but these interpretations vary wildly.

Bayesian deconvolution, inverse problem, and uncertainty quantification most universally refer to using the prior model to regularize the ill-posed inferences from these models. Sometimes this might be used to refer to empirical Bayesian methods and sometimes principled prior elicitation.

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