Question

Let the input-output relationship of a system at time-i be given as (for i = 1, . . . , n) yi = hxi zi let x = [x1, . . . , xn ] t be the random vector corresponding to the input, y = [y1, . . . , yn ] t be the random vector corresponding to the output, and z = [z1, . . . , zn ] t be the random vector corresponding to the noise. we are told that x consists of jointly gaussian random variables with mean vector mx and covariance matrix cxx. we are also told that z contains i.i.d. zero-mean gaussian random variables with variance σ 2 . in addition, we know that h follows a laplacian distribution with fh(h) = 1 2b e − |h| b for some parameter b > 0. we also know that h, x and z are statistically independent. suppose we are given the observations y = y and x = x. the map estimate of h, given these observations is obtained as hˆmap = arg max h fh|y,x h y, x and the ml estimate may be defined as hˆml = arg max h fy|x,h y , x, h

Answer 1

The MAP estimate of h is obtained as hˆmap = arg max h fh|y,x, and the ML estimate is hˆml = arg max h fy|x,h.

Step-by-step explanation:

In the given scenario, the Maximum A Posteriori (MAP) estimate of the parameter h is derived by maximizing the conditional probability distribution fh|y,x, given the observed vectors y and x.

Simultaneously, the Maximum Likelihood (ML) estimate is determined by maximizing the likelihood function fy|x,h with respect to the parameter h. These estimates provide statistical inferences about the parameter h based on the observed data y and x.

Understanding the mathematical expressions for MAP and ML estimates involves the application of probability distributions and statistical independence assumptions among the variables h, x, and z. The Laplacian distribution for h, Gaussian distributions for x and z, and their independence play crucial roles in formulating these estimates.