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Questions tagged [markov-random-field]

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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 ...
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1answer
386 views

On the convergence of Iterated Conditional Modes (ICM) for MAP inference

ICM is very fast but I could not find any references that contain a detailed analysis on its convergence (e.g. rate of convergence). Any suggestions please? Thanks a lot for your help!
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26 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 ...
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1answer
137 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 ...
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1answer
167 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 ...
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13 views

Models for interdependent finite sequences?

I have a large set S of pairs of (short) sequences (, )_i where the first sequence of each pair comes from sequence set A and the second sequence of each pair comes from the sequence set B. Sequences ...
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16 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'...
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1answer
100 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 ...
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21 views

Is the Markov Network (Markov Random Field) property biconditional?

As far as I know, the property of a Markov Random Field is defined as follows: Let $G = (V, E)$ be a Markov Network. Let $X, Y, C \subseteq V$. If every path from a vertex in $X$ to a vertex in $Y$ ...
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1answer
291 views

Transition probabilities for Gibbs Sampling in a Markov Random Field

I am currently reading this paper on Restricted Boltzmann Machines. On page 22, Given a Markov Random Field $\mathbf{X} = (X_1,\ldots,X_N)$ w.r.t a graph $G = (V,E)$ where $V = \{1 \ldots N\}$ and $...
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0answers
32 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 ...
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0answers
36 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 ...
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1answer
26 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 ...
1
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1answer
149 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 ...
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0answers
89 views

Moralized graph factorization from Bayesian network

Given the Bayesian network on the left hand side in the following figure, it shows that the random variable $B$ is dependent on $A$ and $C$, and the Bayesian network $G$ can be factorized as: $P(G) = ...
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1answer
110 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 ...
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0answers
31 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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1answer
133 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 ...
2
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1answer
220 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 ...
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2answers
1k 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 ...
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0answers
258 views

Factorization of a Markov random field

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 \...
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1answer
732 views

derivation of partition function in conditional random fields

When reading the paper of Efficient piecewise training of deep structured models for semantic segmentation, I am confused about the derivation in CRF training (section 6). In specific, I do not know ...
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0answers
530 views

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 ...
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0answers
412 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 ...
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1answer
93 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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1answer
195 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/...
4
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1answer
328 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 ...
3
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1answer
268 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 $M$ ...