September 29, 2016

## Lecture 7 Main Points Once Again

• Marginal probabilities. Compute marginals of variables (given model parameters $$\mathbf{\theta}$$): $$p(x_i\mid \mathbf{\theta})=\sum_{\mathbf{x}': x_i'=x_i}p(\mathbf{x}'\mid \mathbf{\theta}).$$ (or posterior distribution, aka, query probabilities)

• Technique: Variable elimination to avoid the computational complexity that is exponential in dimension

• Why it works
• Use the fact that some factors only involve a small number of variables
• By computing intermediate factors and caching the results, we avoid duplicated calculations
• Q: What if we calculated a particular query probability $$p(x_1\mid x_6)$$, and now we want to calculate $$p(x_4\mid x_3)$$? How to share the work across them.

• A: This motivates the study of more sophisticated graph representation methods, including factor graphs and tree representation of UG.