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We studied in Chapters 29 and 30 the mean‐square error (MSE) criterion in some detail, and applied it to the problem of inferring an unknown (or hidden) variable from the observation of another variable
when
are related by means of a linear regression model or a state‐space model.
The mean-square-error (MSE) criterion (27.17) is one notable example of the Bayesian approach to statistical inference. In the Bayesian approach, both the unknown quantity, , and the observation,
, are treated as random variables and an estimator
for
is sought by minimizing the expected value of some loss function denoted by
. In the previous chapter, we focused exclusively on the quadratic loss
for scalar
. In this chapter, we consider more general loss functions, which will lead to other types of inference solutions such as the mean-absolute error (MAE) and the maximum a-posteriori (MAP) estimators. We will also derive the famed Bayes classifier as a special case when the realizations for
are limited to the discrete values
.
