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GAM model with spatial account via MRF

I am starting to read a little about generalized additive models and how spatial dependencies can be incorporated into the model, I'm reading Generalized Additive Models An Introduction with R from ...
Raheshi Knuwga's user avatar
1 vote
1 answer
97 views

Understanding the Ising Model and finding the MLE

In a binary pairwise MRF, the joint distribution is as follows: \begin{align} p(x\mid\theta) & = \exp\left(\sum_{s \in N} \theta_s x_s + \sum_{(s,t) \in E} \theta_{st} x_s x_t - \Phi(\theta)\right)...
dlu's user avatar
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What type of probabilistic assumption do we make in a MRF if we break a clique by pairwise potentials?

If we have a fully connected MRF with three random variables $a, b, c$, what probablistic assumption would we make if we break the joint potential of the three by pairwise potentials? $$ \phi(a,b,c) = ...
Adrian's user avatar
  • 11
2 votes
1 answer
386 views

Why is computing the partition function expensive?

The joint distribution of a undirected graph can be factorized as a product of potential functions over the maximal cliques of an undirected graph. $$ p(\mathsf{x} \mid \theta) = \frac {1} {Z(\theta)} ...
GaryTheBaddy's user avatar
2 votes
1 answer
111 views

Why are undirected graphical models (MRFs) not represented directly in terms of probability like directed graph models?

I have been reading the Deep Learning Book by Ian Goodfellow, and in that, there is a discussion about graphical models like Bayesian belief networks and Markov Random Fields. Here: One key difference ...
Kunj Mehta's user avatar
1 vote
0 answers
72 views

Why a undirected graph is Markov equivalent to a directed graph iff it is decomposable?

Claim 1. A undirected graph is Markov equivalent to a directed graph iff the undirected graph is decomposable. I am trying to prove Claim 1 and to find a relationship between decomposable and v-...
DLOX's user avatar
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2 votes
1 answer
80 views

Is it always possible to find a joint distribution $p(x,y)$ consistent with the results of both labs?

I am reading "Bayesian Reasoning And Machine Learning" and doing exercise 4.8 and would like to check if the following reasoning is correct. Two research labs work independently on the ...
Slim Shady's user avatar
2 votes
1 answer
99 views

How can I derive the joint distribution for this Markov network?

I am reading Bayesian Reasoning And Machine Learning and I'm not sure how to do exercise 4.6 on p.80. The undirected graph: represents a Markov network with nodes $x1, x2, x3, x4, x5$, counting ...
Slim Shady's user avatar
2 votes
1 answer
141 views

How can I show that these two variables in a Markov network are marginally independent?

I am reading "Bayesian Reasoning And Machine Learning" and I'm doing exercise 4.2 on page 79. This is the exercise: Consider the Markov network $$p(a,b,c)=\phi(a,b)\phi(b,c)$$ Nominally, by ...
Slim Shady's user avatar
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1 answer
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Proof of multivariate distribution using exponential families and Hammersley Clifford Theorem

I'm reading the following seminal paper by Besag http://www2.stat.duke.edu/~scs/Courses/Stat376/Papers/GibbsFieldEst/BesagJRSSB1974.pdf I'm unsure how they prove on page 10 equations 4.4 and 4.5 ...
Pavan Sangha's user avatar
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1 answer
590 views

When and why converting a Bayesian network into a Markov Random Field?

I found many slides and tutorials (e.g., [1,2]) on the probabilistic graphical model introducing the procedure of "converting a Bayesian network (BN) into a Markov random field (MRF) by ...
user1036719's user avatar
1 vote
0 answers
57 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 ...
Pavan Sangha's user avatar
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0 answers
42 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 ...
Shan's user avatar
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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)} ...
carlo__'s user avatar
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133 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 ...
Charlie's user avatar
  • 121
4 votes
1 answer
63 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 ...
Tor's user avatar
  • 205
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1 answer
1k 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, $...
jnez71's user avatar
  • 218
3 votes
1 answer
437 views

Factor graph equivalent to Markov networks

Consider the following potential on three nodes. $$\psi(x_1,x_2,x_3) = f_a(x_1,x_2)f_b(x_2,x_3)f_c(x_1,x_3)$$ represented by the following factor graph: Now the notes claim that we can represent this ...
Pavan Sangha's user avatar
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0 answers
117 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 ...
e. erhan's user avatar
  • 107
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0 answers
61 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 = ...
mdeff's user avatar
  • 148
4 votes
1 answer
152 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 ...
krishnab's user avatar
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1 vote
0 answers
33 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 ...
Mean Difference's user avatar
0 votes
1 answer
516 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 ...
asdf's user avatar
  • 353
0 votes
1 answer
288 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 ...
Merry's user avatar
  • 255
11 votes
1 answer
2k 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. ...
asdf's user avatar
  • 353
1 vote
0 answers
144 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 ...
Yohanes Setiawan's user avatar
1 vote
0 answers
32 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 ...
Sia Rezaei's user avatar
2 votes
0 answers
214 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 ...
Ash's user avatar
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4 votes
0 answers
324 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 ...
Lucidnonsense's user avatar
1 vote
0 answers
122 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.
John Deterious's user avatar
1 vote
0 answers
21 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 ...
mathlover's user avatar
1 vote
0 answers
311 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 ...
groove's user avatar
  • 503
9 votes
1 answer
1k 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 ...
colddie's user avatar
  • 193
1 vote
1 answer
407 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 ...
Chill2Macht's user avatar
  • 6,309
2 votes
1 answer
789 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 ...
Jeremy Lane's user avatar
1 vote
0 answers
64 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://i.sstatic.net/UKh8B.jpg Now if we want to sample P(A|B), how would we do ...
Ambikeya Sangwan's user avatar
1 vote
0 answers
85 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 ...
hanugm's user avatar
  • 205
0 votes
1 answer
151 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 ...
hanugm's user avatar
  • 205
2 votes
1 answer
1k 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 ...
elias's user avatar
  • 21
3 votes
1 answer
341 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 ...
JYY's user avatar
  • 757
1 vote
0 answers
45 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 / ...
Adrian's user avatar
  • 4,374
4 votes
1 answer
258 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 ...
Cauchy's user avatar
  • 93
0 votes
0 answers
487 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 \...
JYY's user avatar
  • 757
3 votes
0 answers
300 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) = ...
JYY's user avatar
  • 757
2 votes
1 answer
675 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 $...
Lycan22's user avatar
  • 73
2 votes
1 answer
370 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 ...
theQman's user avatar
  • 677
2 votes
1 answer
689 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!
f10w's user avatar
  • 213
3 votes
1 answer
2k 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 ...
user785099's user avatar
  • 1,307
2 votes
1 answer
710 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 ...
Tianchen's user avatar
6 votes
2 answers
3k 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 ...
Diego Tsutsumi's user avatar