# Questions tagged [markov-random-field]

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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 ...
1 vote
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### 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)...
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### 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 ...
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1 vote
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### 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-...
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### 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 ...
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### 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 ...
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### 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 ...
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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 ...
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1 vote
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 ...
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1 vote
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 ...
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### 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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### 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 ...
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### 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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### 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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### 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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### 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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