Segment 21. Marginalize or Condition Uninteresting Fitted Parameters

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To Calculate

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

2. Same, but marginalize over Failed to parse (unknown error): b_2 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.