A common problem in signal processing is to recover a signal x from measurements y. Examples include image denoising and deblurring, speech enhancement and dereverberation, target tracking and more. Situations of this type are often tackled by employing Bayesian estimation techniques such as the minimum mean-squared error (MMSE), linear MMSE (LMMSE) and maximum a posteriori (MAP) estimators. The Bayesian framework relies on the assumption that x is a random quantity drawn from some prior probability density function f_X(x) and that the statistical relation between y and x is characterized by a likelihood function f_Y|X(y|x).

The prior f_X(x) can typically be learned from a set of examples of clean signals. Indeed, a large variety of databases of all kinds of signals are available online, including facial images, fingerprints, iris scans and speech signals to name a few. The likelihood f_Y|X(y|x), on the other hand, is associated with the degradation mechanism in a specific application and thus cannot be learned from databases of this sort. One possibility for obtaining f_Y|X(y|x) is to assume a known degradation model such as additive white Gaussian noise. However, in many situations this assumption is over-simplistic since the degradation includes complicated effects, which are hard to model and sometimes not even known. Nonlinear distortion in CCD sensors, unknown blur and signal dependent noise are a few examples of such phenomena. An alternative approach for obtaining f_Y|X(y|x) is to learn it by collecting a paired set of examples of clean and degraded signals. Unfortunately, constructing such a database requires a complicated experimental setting in which our sensor is co-calibrated with some high-quality sensor, and is therefore usually impractical. Specifically, it is typically quite simple to obtain a set of clean signals from some a high-grade sensor (or from an existing database). Similarly, it is also easy to collect a set of degraded signals taken with our low-grade sensor. But these two sets are unpaired.

How can we estimate x from y without knowing the degradation model? The answer is simple – employ an instrument z. An instrument is a random quantity whose statistical relations with x and y are known or can be learned from examples. Consider, for instance, the task of enhancing a video sequence y of a speaker taken from a low-quality cellular-phone camera. In this case we can use the audio z as an instrument. Clearly, we can collect paired examples {y_n,z_n} of the noisy video and its associated audio (taken with the cellular-phone), as well as paired examples {x_n,z_n} of clean video sequences with their audio (taken from a high-quality camcorder), as schematically shown below.

The training sets in this situation can be used to learn the densities f_XZ(x,y) and f_YZ(y,z) but are generally insufficient for determining f_Y|X(y|x).

The difference between the MSE of an estimator and the lowest possible MSE that could be achieved if f_Y|X(y|x) was known, is called regret. The regret attained by any estimation strategy depends on the unknown likelihood f_Y|X(y|x). Our approach in this work is to design an estimator whose regret for the worst-case likelihood, which is consistent with our knowledge of f_XY(x,y) and f_YZ(y,z), is minimal. We call this technique the partial knowledge minimax regret estimator.

• T. Michaeli and Y. C. Eldar, "A Minimax approach to Bayesian estimation with partial knowledge of the observation model", IEEE Int. Conf. on Acoustics, Speech and Signal Processing (ICASSP 2010).

• The minimax regret estimator is implemented in the Matlab function MinimaxRegretEstimator.m.

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