## Monday, April 9, 2012

### Normalization

I'm sorry I wasn't able to be there in class today. However, Coco reports that you all made good progress working on Class Activity 1. She also reports that many of you struggled with the normalization part of Problem (4). Assuming this is where the problem was, allow me to help everyone along.

Problem (4) states:

Without properly normalizing things, you will end up with a proportionality of the form:

$\large p(\mu\,|\,\{A\}) \propto p(\{A\}\,|\,\mu)\ p(\mu)$

In order for this to be an equation, you have to normalize it. You might want to convince yourself that the quantity on the right hand side is not normalized by integrating over all values of \mu. In fact, dividing by this integrated quantity provides the normalization constant:

$\large p(\mu\,|\,\{A\}) =\frac{p(\{A\}\,|\,\mu)\ p(\mu)}{\int_{-\infty}^{\infty} p(\{A\}\,|\,\mu)\ p(\mu) \rm{d}\mu}$

The denominator is also known as the "evidence":

$\large p(\{A\}) =\int_{-\infty}^{\infty} p(\{A\}\,|\,\mu)\ p(\mu) \rm{d}\mu$

or the probability of the data given the model, with \mu marginalized out.

The actual value of the integral can be expressed analytically, or you could just do it numerically. Or you can use WolframAlpha. But whatever you do, don't get too hung up on this! :)