Argonauts:Markov Random Fields and Bayes
From Wasteland
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Sat 11 March 2006
Spatio-temporal coordinates
WMVL, 11.00am to 1.00pm (exactly)
Attendees (in alphabetical order)
- Ramón Casero Cañas.
- Niranjan Joshi.
- Ingmar Posner.
Minutes
Our intention in this meeting was to put together what we know of Markov Random Fields and of Bayes theorem and Maximum a posteriori.
We decided to go through the 143 slides of
M.A.T. Figueiredo. Bayesian Methods and Markov Random Fields. Department of Electrical and Computer Engineering. Instituto Superior Tecnico. Lisboa, Portugal.
and of course we didn't have enough time.
We have a quantity x that we want to estimate from some observations s. Bayes theorem tells us that the a posteriori conditional probability of x having observed s is
P(x|s) = P(s|x) * P(x) / P(s)
We can define an error function called the a posteriori expected loss
E{L(x) | s}
from a certain error metric. If we minimize E{L(x) | s}, then we obtain the optimal estimate of x in some sense. For example, the MAP estimate is the limit case of using a "0/1 loss function" (it's a mode estimate). For a quadratic loss function, we obtain the posterior mean estimate.
If we assume Gaussian observations with a Gaussian prior, we get to what we have seen of Kalman Filters. If the prior is uniform, then we have the Maximum Likelihood estimate (ML).
Nice, isn't it? Many of the things we have seen previously smoothly come together!
The introduction to MRFs starts in slide 65, and at this point it was clear that we were not going to be able to finish the whole thing, even if we skipped a big chunk of Bayesian theory. Figueiredo describes MRFs as a convenient tool to write priors for image, which is what we expected when we got into this topic.
From Local Markovianity follows that the pdf of intensity values in each pixel depends only on the intensity values of neighbouring pixels. Local neighbourhoods are described using graphs.
The Hammersley-Cliford theorem enables us to write the prior as a Gibbs distribution
p(x) ~ exp( sum_i V_i )
where V_i are clique potencials, i.e. some distance metric between a pixel and its neighbours.
Gauss-Markov Random Fields (GMRFs) arise when potencials follow an auto-model (we have only cliques with pair-wise terms, i.e. each potencial is just some distance between one pixel and one neighbour). In the end this just means that the pontecials in the Gibbs distribution can be written as
V_i ~ x^T A x
where A is the inverse of the covariance matrix. The catch is, of course, how to define the covariance matrix between pixels. This is were we run out of time. Niranjan knows loads about this, having been working with it for his thesis, and he hinted that this is not trivial at all.
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