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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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Using mcmc
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

Articulate Probabilities

[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

To get one answer

(if you want)

Gives you one answer


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

Good, Bad

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
Line Fitting (again) iterations can be very low complexity

Needs Latex

Observed Data

Actual Line

Initial Guess


Typical results
Typical Results iterations can be very low complexity

See Matlab demo

Nice Convergence

because we can draw samples directly


Typical results1
Typical Results iterations can be very low complexity

c

m

var_e


Watch out
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
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
Equaliser Problem iterations can be very low complexity

Now more latex


Direct numerical sampling
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 equaliser1 m
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 equaliser2 m using filter bank system choices
Gibbs Sampler II iterations can be very low complexity(equaliser2.m)Using Filter Bank (system choices)

30 filters


Samples from filter bank
Samples from filter bank iterations can be very low complexity


Samples of signal parameter
Samples of signal parameter iterations can be very low complexity


System estimate
System Estimate iterations can be very low complexity


Equalised signal
Equalised signal iterations can be very low complexity


Lessons
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


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