Segment 21. Marginalize or Condition Uninteresting Fitted Parameters

From Computational Statistics (CSE383M and CS395T)
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Problems

To Calculate

1. Consider a 2-dimensional multivariate normal distribution of the random variable <math>(b_1,b_2)</math> with 2-vector mean <math>(\mu_1,\mu_2)</math> and 2x2 matrix covariance <math>\Sigma</math>. What is the distribution of <math>b_1</math> given that <math>b_2</math> has the particular value <math>b_c</math>? In particular, what is the mean and standard deviation of the conditional distribution of <math>b_1</math>? (Hint, either see Wikipedia "Multivariate normal distribution" for the general case, or else just work out this special case.)

2. Same, but marginalize over <math>b_2</math> instead of conditioning on it.

To Think About

1. Why should it be called the Fisher Information Matrix? What does it have to do with "information"?

2. Go read (e.g., in Wikipedia or elsewhere) about the "Cramer-Rao bound" and be prepared to explain what it is, and what it has to do with the Fisher Information Matrix.

Class Activity

Today we'll do Find the Volcano.