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# Random Vectors

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Vectors  ::  Covariance  ::  Functions  ::  Application  ::  Markov Model

## An Application: MMSE Prediction

Suppose we have a random sequence , and we observe the first of them:

Given this data, we want to predict the value of .

Our estimate of will be denoted as . Clearly, it could be a function of all the observed data:

for some function . One thing we could try is to minimize the average of . That is we would like to solve

It is easy to see (HW!) that the best such is

That is, the best estimator (in a minimum mean-squared error sense) is the conditional expectation!

Now let us take a specific distribution. Suppose , and partition according to

Given , the variable is where

where

and

So is the conditional mean that we want and is the variance of the conditional distribution,

This is the minimum mean-squared error (MMSE).

Notationally, write

Then

This is just a digital filter!

We can also show that , so that incorporating information from measurements decreases our uncertainty.

Note: For a Gaussian r.v., the MMSE estimator is linear .

Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, May 31). Random Vectors. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Stochastic_Processes/lec3_4.html. This work is licensed under a Creative Commons License