Sample topics for a project:

HMMs:
  Structured HMMs: e.g. factorial HMMs, dynamic Dayesian networks
   Spectral methods for learning HMMs

Dynamic programming:
  Belief propagation
  The inside-outside algorithm for probabilistic context free grammars
  Belman's equation (e.g. for portfolio optimization, optimal control)


EM:
  Online EM algorithms
  Variational bayes

Monte Carlo methods:
  Particle filtering
  Annealed importance sampling
  MCMC for Dirichlet process mixtures
  Parallel MCMC
  Approximate MCMC (`MCMC on a budget')
  Perfect sampling/coupling from the past

Optimization methods:
  Bayesian optimization
  Convex optimization (lots of topics)
  Combinatorial optimization (lots of topics)

Misc ideas:
  Random forests, 
  Lasso, 
  Boosting,