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Using MCMC - PowerPoint PPT Presentation

Using MCMC. Separating MCMC from Bayesian Inference? Line fitting revisited A toy equaliser problem Some lessons A problem in film restoration/retouching. Articulate Probabilities [Bayesian Inference]. Try to see if you can integrate out nuisances. Derive the Posterior.

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PowerPoint Slideshow about 'Using MCMC' - edison

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Presentation Transcript

• Separating MCMC from Bayesian Inference?

• Line fitting revisited

• A toy equaliser problem

• Some lessons

• A problem in film restoration/retouching

[Bayesian Inference]

Try to see if you can

integrate out nuisances

Derive the Posterior

MCMC is (just) a tool

Choose a Model

Identify Parameters

Need better model

If solution not ok

Use MCMC

Solve Deterministically

Direct, CG, Steepest Descent etc

Manipulate Random Samples

(if you want)

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

Can give local minimum for non-linear problems

Others

MCMC

Ugly iterations can be very low complexity

• 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) iterations can be very low complexity

Needs Latex

Observed Data

Actual Line

Initial Guess

Typical Results iterations can be very low complexity

See Matlab demo

Nice Convergence

because we can draw samples directly

Typical Results iterations can be very low complexity

c

m

var_e

Watch out iterations can be very low complexity

• 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 iterations can be very low complexity

Noise

Signal

2nd Order

All pole System

Rec’d Signal

Identify the system coefficients

AND recover the original signal

Comms, Deblurring, Overshoot Cancellation

Equaliser Problem iterations can be very low complexity

Now more latex

Direct numerical sampling iterations can be very low complexity

P(1) = 0.3, p(2) = 0.25,

P(3) = 0.2, p(4) = 0.25

0

1

0.3

2

0.25

3

0.2

0.75

4

0.25

1

Number line

interpretation

71 points evaluated

Gibbs sampler 1 iterations can be very low complexity(equaliser1.m)

Back to Latex

Typical

Estimated System

Actual System

X 20 !

300 iterations

Gibbs Sampler II iterations can be very low complexity(equaliser2.m)Using Filter Bank (system choices)

30 filters

Samples from filter bank iterations can be very low complexity

Samples of signal parameter iterations can be very low complexity

System Estimate iterations can be very low complexity

Equalised signal iterations can be very low complexity

Lessons iterations can be very low complexity

• 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