# EXERCISES

- 1. What is the difference between clustering using a Gaussian mixture model and К-means using the Malhalanobis distance?
- 2. Derive the update equation for the mixing parameter for the two component mixture of sequence model.
*(Hint:*Because I’ve written the likelihood using one parameter for the two components, you don’t have to bother with Lagrange multipliers. Just take logs, expectations, and differentiate directly.) - 3. Show that the expectation for a Bernoulli variable (or any 0 or 1 variable) is just the probability of the positive outcome. In other words: E[X] =
*P(X*= 1) if*X*is {0, 1}.*(Hint:*Just plug in the values for the Bernoulli distribution into the standard formula for the expectation E[] [given in Chapter 2].) - 4. How many parameters are there in a mixture of Gaussians model with diagonal covariances (as a function of
*К*and*d)?* - 5. Show that the discrete probability model
*p(Xf)*used for sequence by MEME is a valid probability distribution, such that the sum over all sequences of length*w*is 1. - 6. Notice that the causality in the Holmes and Bruno model is different from the biological causality: biologically, the binding to DNA would recruit the regulator, which would then cause expression changes. Draw the graphical model representation of this model and write out the formula for the joint distribution including the dependence structure.
- 7. Draw a graphical models representation of the model where DNA affects RNA and both DNA and RNA affect a disease, but the disease doesn’t affect either DNA or RNA. Draw the graphical model representation of this model and write out the formula for the joint distribution including the dependence structure.
- 8. Hidden variables are widely used in evolutionary modeling to represent unobserved data from ancestral species that are not observed. For example, if we use
*A*to represent the unobserved ancestral data, and*X*and*Y*to represent observed data from two extant, descendants of*A*, the joint probability of the observed data can be written

*P(X,Y*) = ^^ *P(X* | *A)P(Y* | *A)P(A).* Draw a graphical models diagram for this probabilistic model. What is *P(A)* in this formula?

9. Explain the assumption about evolution along each branch being made in the model in question 8.