# Is it possible to reformulate a Kalman Filter as a Gaussian Markov Random Field?

The generic formulation of a KF uses a set of transitition equations, while the GMRF is typically specified through its mean and precision. However, a simple KF involves Gaussianity and Markov conditioning, so there must be some connexion with GMRFs. Some GMRFs can see the future, while a KF cannot, so there must be a specificity in the formulation of the precision matrix. So, is it possible to specify a precision matrix for a KF ?

Edit: a GMRF is a Multi-Variate Gaussian distribution whose precision (the inverse of the covariance) has many zeroes. In addition to obvious nice computational properties, it induces conditional independence between two components (resp groups of components) who have a zero on their shared coefficient (resp blocks of components). If you want to know more, look at this gem: https://www.taylorfrancis.com/books/mono/10.1201/9780203492024/gaussian-markov-random-fields-havard-rue-leonhard-held. A really useful reformulation of the GMRF is the Cholesky facotrization of the precision matrix, whose rows give the conditional expectation and variance of one component knowing all previous components. Like the precision matrix, its Cholesky factor is sparse. This is where my question comes from. We have Gaussianity in both cases. We have a recursive conditional formulation. We have sparsity. That's a lot. However, the typical formulation of the GMRF will give the distribution of the latent variables knowing all the observations, while the KF gives the distribution of a latent variable knowing only the previous observations. So I was wondering if the KF could be reformulated as a GMRF, with a special precision / Cholesky precision structure. It's quite a tricky question and I don't expect someone who is new with GMRFs to find an answer, but I just tossed a bottle into the sea. Thanks for reading anyway !

• Kalman filters certainly can see the future, in the sense that you can predict future states from the current state and the model. Try searching on "Kalman predictor". Commented Oct 12, 2023 at 17:07
• Looking at the Wikipedia description for a Markov random field, it looks like the MRF is a static description of a multivariate probability problem, while a Kalman filter describes one that evolves over some variable -- i.e., time. I'm not sure, though, or I'd make this an answer. Basically, if you can transform a GMRF description into a Kalman filter -- yes. If you can't -- no. Commented Oct 12, 2023 at 21:40
• I should have thought of this earlier -- can you edit your question with a citation that describes a GMRF? It's not a common enough thing to have made its way into Wikipedia, and their description of the Markof random field is pretty sketchy. Commented Oct 12, 2023 at 21:47
• of course @TimWescott
– SCS
Commented Oct 13, 2023 at 8:19
• Hello there welcome here. Unless somebody comes and proves me and Tim Wescott wrong I believe GMRF to be a niche subject. Also the reference you gave is kept behind a subscription which I am not necesseraly interested in and anyway it's 280 pages long! So I suggest you do one of two things : if you believe we can get on page easily you describe to us what GMRF are and we'll try to answer. If you believe it requires deep knowledge in statistics you might as well take a chance on crossvalidated which is the stack exchange specialized on statistics. Commented Oct 13, 2023 at 13:20