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Questions tagged [markov-random-field]

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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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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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1answer
261 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 $M$ ...
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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 ...
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1answer
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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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1answer
105 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
272 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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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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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
183 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/...
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1answer
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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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1answer
642 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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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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0answers
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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
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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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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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1answer
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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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1answer
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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 ...
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1answer
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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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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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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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Conditional Random Field for Hyperspectral Image Classification

How can I separate training and test sets in a hyperspectral image and apply Conditional Random Field (CRF) for pixel classification? If I choose pixels randomly, some of the neighboring pixels of a ...
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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
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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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1answer
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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!