Using MCMC Separating MCMC from Bayesian Inference? Line fitting revisited A toy equaliser problem Some lessons A problem in film restoration/retouching

MCMC is (just) a tool Choose a ModelIdentify Parameters Articulate Probabilities [Bayesian Inference] Try to see if you can integrate out nuisances Derive the Posterior Solve Deterministically Direct, CG, Steepest Descent etc Use MCMC Manipulate Random Samples To get one answer (if you want) Gives you one answer Need better model If solution not ok

Good, Bad Can always design single parameter-at-a-time schemes. So iterations can be very low complexity Simple iterations = long convergence Gives you a picture of alternate answers Do you really need alternate answers? Will always allow you to get to “best” solution Iterations can be high complexity? Convergence can be rapid (e.g. CG) for well defined problems Gives you just one answer Can give local minimum for non-linear problems MCMC Others

Ugly To solve your problem you need a good model MCMC is not really going to help you if you have the wrong model MCMC suited to BIG problems: but what is BIG really? E.g. Exhaustive search for motion estimation is possible in real time (TV rates) in hardware: why bother with other things? (an exaggeration … but interesting nevertheless)

Line Fitting (again) Needs Latex Observed Data Actual Line Initial Guess

Typical Results Nice Convergence because we can draw samples directly See Matlab demo

Typical Results c m var_e

Watch out All random number generators are not created equal (See NR) Harder problems require longer runs (of course) Sometimes hard to get all bugs out because its all a random search anyway

Blind (?) Equalisation Signal2 nd Order All pole System Noise Rec’d Signal Identify the system coefficients AND recover the original signal Comms, Deblurring, Overshoot Cancellation

Equaliser Problem Now more latex

Direct numerical sampling Number line interpretation P(1) = 0.3, p(2) = 0.25, P(3) = 0.2, p(4) = points evaluated

Gibbs sampler 1 (equaliser1.m) Typical Estimated System Actual System X 20 ! 300 iterations Back to Latex

Gibbs Sampler II (equaliser2.m) Using Filter Bank (system choices) 30 filters

Samples from filter bank

Samples of signal parameter

System Estimate

Equalised signal

Lessons Gibbs sampler takes big problems and breaks them into lots of small ones Spotting the functional form of a known p.d.f. is a useful skill. Books help. If all else fails, can always sample directly MCMC does not necessarily solve your problem. Good priors, better models are still important Deterministic/Stochastic Hybrid mix is v. useful