1. The 2 variants of the binomial approximation formula for partial partition function in Ising models with fixed edge potential and weak node potentials
  2. Turns out, MAP makes the same number of mistakes as the trivial estimator! It still achieves lower error rate because the locations in which it makes these errors bear lesser mass compared to the trivial estimator.
  3. Network effect is seen in moderate SNR region. Not much of a difference in the low and the High SNR regimes...

MSE vs topology for GMRFs

Closed form expressions for the MSE for stylized topologies such as complete graph, star and cycle seem very doable.. and so are the error exponents

First order approximation for MAP detector for disparate weak external fields.

Finally, we have a closed form expression for the MAP detector. Its a the first order approximation valid for weakish edge and node potentials...
Icing on the cake: Decision statistic does not need the entire adjacency matrix, Just the first order degree information.

Some candidates for open data spatial -MRFs

Still scouting around for meaningful open data spatial -MRFs ... Just collected a few good candidates.

Keeping up with the GMRFs

Turns out the family of GMRFs is quite a motley one. Researchers have specified these models in varied fashion across the body of literature. Here are some prime examples:

  1. Intrinsic (Improper)
  2. Linear filters specification as in the perturb-and-MAP paper.
  3. Via the covariance matrix
  4. Generalized CAR
  5. Besag's classical zero-mean CAR
  6. AR/SAR.
The CAR vs SAR dilemma was pretty informational as Katt Williams put it. The bottom line seems to be that SAR models are sorta more well suited to ML estimation (but not so much for MCMC fitting). The
hierarchical conditional representation of CAR helps it trump here. 
*Must read more*

Some chump change MATLAB code, model comparisons

This post, I reckon is mostly for book keeping. To remind me the not-so-subtle difference in the MSE when the two seemingly cross-compatible models are switched.
Nice 'data science' lesson this ...

Chain graph - Full graph -MSE

Hmm.. time to ponder. MSE is not playing nice as a metric. Need to head back to the drawing board. The difference in MSE seen when the topology is switched from full network (complete graph) to chain should be atleast a paradigm shift.

Emergency visits in CA counties

Let's assume that the underlying graph is the adjacency inter-county graph of the counties of CA.
Let the county label be +1 if its emergency visit rate is above the state average and -1 if otherwise.
Does this constitute an Ising model?
Approach: p-value driven

GMRF based imputation in Average emergency visits

Lets look at the California dataset of average emergency visits. Can we throw a GMRF+SVR regression model at it and impute missing values?
If so, how good are the results?

Bhattacharya co-efficient + GMRF

Bhattacharya co-efficient and gradient evaluation: Some hand calculations here ...