# Questions tagged [markov-random-field]

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### What's the difference between a Markov Random Field and a Conditional Random Field?

If I fix the values of the observed nodes of an MRF, does it become a CRF?
2k views

### Markov Random Fields vs Hidden Markov Model

I'm kinda new to these topics, I wanted to know if there are any relations between those two topics, Markov Random Fields and Hidden Markov Models (Markov Chains). I feel like they are completely ...
403 views

### Markov Random Field Non-Positive Distribution

The joint distribution in a Markov Network can be represented as: $P(X=x) = \frac{1}{Z}\phi_k(x_k)$ where $\phi_k$ represents the $k^{th}$ factor. While reading Improving Markov Network Structure ...
179 views

### use MCMC posterior as prior for future inference

Would you kindly let me know how to use the estimated posterior distribution as the prior of another Bayesian update? Or even use that in an iterative manner, e.g. in my case the posterior is updated ...
289 views

### Sampling a random binary matrix with “Gaussian” probability distribution

Let $A_{ij}$ be a $n\times n$ random binary matrix with probability mass function $P(A)$ given by $$\log P(A)=-\frac 12 \mathrm{tr}\left[\left(A-M\right)^TV\left(A-M\right)\right] + C,$$ where $M$ ...
145 views

### Markov random field potentials

Consider a pairwise Markov random field, for any two neighbours $A$ and $B$, is it correct to use any function to describe the relationship between them? Is there any constraint or any condition that ...
230 views

### Confusion regarding terminology related to the junction tree algorithm

As far as I understand, the "junction tree algorithm" is a general inference framework which roughly consists of the four steps 1) triangulate, 2) construct junction tree, 3) propagate probabilities/...
165 views

### Mean field theory and neural networks

Mean field algorithm has been proposed to be used in combination with convolutional networks and recursive neural networks. What is the purpose of doing this? Is the goal to estimate a probability ...
243 views

### Understanding the distribution of the Spike & Slab Restricted Boltzmann Machine (ssRBM)

The ssRBM is described as a way to model mean and covariance using Restricted Boltzmann Machines. I'm reading the paper that introduced the spike and slab restricted boltzmann machine. I have yet do ...
337 views

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### How to express Bayesian Network or Markov Random Field using deep learning

Bayesian Nework and Makov random field are instances of general probabilistic graphical model. Is it possible to express Bayesian Network or Markov Random Field using deep learning? or in general to ...
492 views

### How to implement a Gaussian Markov Random field (GMRF) model in R?

I would like to model a (conditional) GMRF using a linear mixed effects model without having grid Data but only a neighbourhood matrix $W$. My model is given by $$Y=X\beta+ \epsilon$$ and the error ...
59 views

### Is every distribution factorizable by an MRF also factorizable via a Bayesian network? And vice versa?

This has probably been asked before, so if it has please provide a link to the original question and close this as a duplicate -- I was not able to find the original question myself. Question: Let's ...
271 views

### How does Markov random field (bs=mrf) in mgvc gam handle repeated measures on the spatial units?

I am attempting a spatio-temporal model in mgcv gam. I am using a factor smooth to define each of 27 areal units in a shapefile ("id") as subjects (essentially) which have undergone 23 repeated ...
206 views

### MRF definition: not all cliques are required to have factors?

I'm reading the notes here. The formal definiton states A Markov Random Field (MRF) is a probability distribution $p$ over variables $x_1,\ldots,x_n$ defined by an undirected graph $G$ in which nodes ...
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### Markov Random Field Implementation

Honestly, I'm having trouble understanding the general steps taken to actually use a Markov Random Field (MRF) in practice. Here's my current thinking on what I would have to do in practice. Specify ...
64 views

### Relation between Gaussian Processes and Gaussian Markov Random Fields

As a non expert in the field, I am relating Gaussian Processes (GP) and Gaussian Markov Random Fields (GMRF). I might just be confused by the fact that different resources use different formalism. ...
39 views

### Edited — Normalizing Constant Z in Markov Random Fields [closed]

I want to implement this paper in MATLAB. This is the formula: $$P(L)=\prod_{s \in S} \frac{1}{Z} exp(- \sum_{c \in N^w(s)}V_c(L_s))$$ However, I confused to compute $Z$ (normalizing constant) as ...
21 views

### Why the nodes in a Boltzmann machine need to be sampled one at a time?

Typically, we use Gibbs sampling to update (or generate samples from) energy based models. This means we update each node while keeping its markov blanket constant. Why can't we update/sample all ...
15 views

### Extension of Potts model with non-constant interactions?

Is there any work that extends (allows) Potts model to have non-constant interactions between the lattice points? Specifically, the interaction matrix is a symmetric matrix that can have both positive ...
34 views

### Discriminative Models with Class Priors

In discriminative models, we model $p(Y|X)$ directly while in generative models we model $p(X|Y)p(Y)$ where $X$ is the input and $Y$ is the output variable. I am confused when the parameters and ...
47 views

### How exactly does Gibbs sampling work in Markov Networks?

I was going through the Probabilistic Graphical Modelling course by Stanford and they used a network such as this one-https://imgur.com/gallery/k0C8FY2 Now if we want to sample P(A|B), how would we ...
34 views

### Are self loops allowed in Markov networks?

I am studying about Markov networks from Probabilistic Graphical Models: Principles and Techniques Book by Daphne Koller and Nir Friedman. In Bayesian networks, it is clear that, it is a directed ...
32 views

### References on simulating a raster with spatial dependence

I've simulated an N-by-N raster in the following way: define a set $S$ containing a finite number $|S| = K$ of possible raster values (in my simulation, $K=3$ and the elements of $S$ are land uses / ...
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### Replicating an experiment on GMRF (Gaussian Markov Random Field)

I am trying to understand an experiment from this paper, specifically Section 5.2. In the paper, they propose a new algorithm for computing the log-determinant of sparse matrices, and in section 5 ...
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### Equivalence between directed and undirected graph?

I am confused over something that may have an obvious explanation I am missing. In Koller's Probablistic Graphical models textbook, page 945, it is said that a Markov network $A-B-C$ is equivalent ...
255 views

### Examples of applications of Markov random fields to data with a small number of variables

I am learning about some of the common applications of Markov random fields (a.k.a. undirected graphical models) to data science. A common feature of many applications I have read about is that the ...
28 views

### How maximal clique parameterization obscures original structure

While reading the chapter on Markov networks, I came across the following statement: Although it can be used without loss of generality, the parameterization using maximal clique potentials ...
96 views

### Derive data likelihood for conditional probability for autologistic model

The log conditional probability for the autologistic model is $\log\Pr(y_i\mid \{y_j : j \neq i\}) = \alpha_iy_i + \sum_j^N\theta_{ij}y_iy_j - \log(1 + \exp(a_i + \sum_j^N\theta_{ij}y_j))$ From ...
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### Difference between CRF and Fully Connected CRF?

Can anyone explain me the difference between Conditional Random Fields and Fully Connected Conditional Random Fields for semantic segmentation? I only understand so far, that with CRF you try to use ...
14 views

### Where is the “Markov property” in Markov Random Fields? Are Markov Random Fields stochastic processes at all?

I am learning about Markom Random Fields. As I understand them they are constituted of some Random variables $X_{i}$ and some potential functions $\phi(X_s)$ where $X_s$ is a set of some random ...
28 views

### Why do we have to convert Bayes' net to MRF before applying Belief propagation?

is that even correct in the first place? if yes, then why? I've seen articles talking about inference in Bayes' nets, and I've seen others talking about conversion. I don't have the full picture.
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### Hidden markov random field

I'm working on HMRF in wireless traffic characterization. Please can someone help? If I have 10x10 matrix how can I apply HMRF on this matrix regarding classification of wireless traffic?
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### Markov network, estimating unknown variable with multiple observations

I have this hidden markov model/network with four unknown variables $y_{1:4}$ with the discrete domain $(0,1)$ and four known observations $y^{obs}_{1:4}$ and a potential function $\phi(x_i,x_j)$.  ...
Consider the Markov random field in the following figure, some literature and textbooks say that the MRF $G$ can be factorized as \$P_1(G) = \phi_1(A,B) \times \phi_2(A,C) \times \phi_3(C,D) \times \...