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Questions tagged [markov-random-field]

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25 views

Approximate inference and the Hammersley-Clifford theorem

I recently submitted a manuscript to a journal in which we attempted to resolve the data association problem inherent in multi-object tracking by way of loopy belief propagation (LBP). We constructed ...
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25 views

log trick on message passing in factor graphs

I'm reading Barbers book on Bayesian reasoning and Machine learning http://web4.cs.ucl.ac.uk/staff/D.Barber/textbook/200620.pdf page 90 To give context this is a proof of using the log trick for the ...
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16 views

Why do we need unary terms in Ising model (pairwise Markov random field)?

Ising model contains both $\phi(i,j)$ and $\phi(i)$. For example, consider a Markov random field with only two nodes $i$ and $j$, if $P=\phi(i,j) * \phi(i) * \phi(j)$, then we can also write $P=\phi(i,...
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13 views

Markov networks which are disconnected

I'm reading Kollers book on PGMs. Some of her examples to show the breaking down of theorems around independencies for non positive distributions involve either empty Markov networks or very sparse ...
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37 views

Re-generate the exact underlying data from an exact MRF model or any other PGMs

I was wondering if there exist a way to re-generate the actual underlying data (not a sample!) from a given exactly learned MRF. In other words, lets say I have a discrete factorised joint ...
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20 views

I-map construction for non positive distribution

I'm reading Kollers book on PGMs and i'm reading this example about the local independence i-map construction for a non positive distribution. Example (As requested to be written out): Consider a ...
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110 views

Weighted adjacency interpretation and visualisation in mixed graphical models

I was wondering if someone could help me understand a bit more about interpretation and visualisation of parameters in mixed graphical models estimated via neighbourhood selection with generalized ...
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25 views

Learning in Markov Random fields

I have a question about learning in Markov random fields. According to the explanation https://ermongroup.github.io/cs228-notes/learning/undirected/ we have $P(x_{1},\ldots, x_{n}) = \frac{1}{Z(\psi)} ...
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15 views

Local independence vs global independence in markov network

I am having a hard time understanding the basic differences between the local independence and global independence of a markov network. Please help me illustrate with a graph or any example
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37 views

Most likely assignment in MRF with a global constraint

I have a MRF where all potentials are on pairs and all variables are binary. I want to find the most likely assignment of variables, under the constraint of at most $k$ variables being 1. Can this be ...
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46 views

Are CRF/MRF/GRF still used widely in computer vision?

I've tried to find recent (the year 2020) popular works that use Markov/Gibbs/Conditional Random Fields. My approach was: go to Google Scholar and find the works, citing a few relevant works on this ...
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1answer
44 views

Probability of at least one success in a long string of connected events

I have N events (i from 1 to N), each with an estimated probability of success, p(i). If all my events were independent I'd be able to calculate the probability of at least one success as (1 - product ...
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1answer
77 views

How can a random variable be independent of a member of its minimal Markov blanket?

Consider the following Bayes network of random variables on some probability space: The local Markov property asserts that any variable is independent of its non-descendants given its parents. Here, $...
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1answer
39 views

Factor graph equivalent to markov networks

Consider the following potential on three nodes. represented by the following factor graph. Now the notes claim that we can represent this factor graph as both a Bayesian network and a Markov ...
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38 views

To estimate a Markov random field, do we have to assume a DAG generated the data?

Even if I believe that I understood the Markov Random Fields and DAGs separately, I encountered a question that I have written in the title and cannot come up with a clear-cut answer. Can you help me ...
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53 views

How does mgcv deal with sum-to-zero constraints for intrinsic Gaussian Markov Random fields?

mgcv states that splines are constrained to sum to zero over the observations. This is to make the model identifiable. Intrinsic Gaussian Markov Random Fields (...
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30 views

Decomposition of a Gaussian Markov random field in independent subfields

A zero-mean GMRF (i.e., a multivariate normal distribution whose precision matrix is sparse) with precision $Q \in \mathbb{R}^{n \times n}$ and covariance $\Sigma = Q^{-1}$ is eigendecomposed as $Q = ...
2
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1answer
57 views

Do variational approximations capture the flow of influence or "conditional independence" relationships in graphical models?

Probabilistic Graphical Models (PGMs) are used to model all sorts of complex decision processes, such as medical diagnoses or robot positions, etc. In common machine learning textbooks, like ...
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21 views

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 ...
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1answer
195 views

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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1answer
46 views

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 ...
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513 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. ...
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72 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 ...
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22 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 ...
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73 views

Why is it difficult to sample from Energy Based Models?

I am trying to understand the following claim which is made in the Deep learning book by Goodfellow et. al about a toy energy-based model (with the apparent motivation of introducing Markov Chain ...
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117 views

Graphical model of the Gaussian mixture: where is n?

TL;DR: Where are the occupation numbers in the Graphical model of the GMM? I am implementing a Finite (to be adapted to infinite later) Gaussian Mixture Model. I am using the Gibbs sampler-ready ...
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62 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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19 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 ...
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0answers
58 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 ...
6
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1answer
433 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
81 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 ...
2
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1answer
417 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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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 ...
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41 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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1answer
49 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 ...
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1answer
614 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 ...
3
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1answer
202 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
35 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 / ...
3
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1answer
208 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 ...
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379 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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156 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) = ...
2
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1answer
361 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 $...
2
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1answer
260 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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1answer
590 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!
2
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1answer
1k 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
612 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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2answers
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 ...
2
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0answers
560 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 ...
2
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1answer
263 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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1answer
98 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 ...