Talk:Markov random field

message: Needs much more work - discussion of inference, the Hammersley-Clifford Theorem, links, etc. Will add more. Revise away. -- Soultaco, 22:50 24 Dec 2004 (UTC)
Could someone please elaborate on this sentence in the intro?: "It can represent certain dependencies that a Bayesian network cannot (such as cyclic dependencies); on the other hand, it can't represent certain dependencies that a Bayesian network can (such as induced dependencies)."  —Preceding unsigned comment added by 18.243.2.30 (talk) 09:59, 2 December 2007 (UTC)
thread_title: Needs much more work

message: From the current version of the article:
*What's the relationship between the potential function and the Gibbs function?
* a set of potential functions \Phi = \phi_{c_i} : A_{c_i} \rightarrow R^+, where each c_i = \lbrace i_1, i_2, ... i_{|c_i|} \rbrace  represents a clique in G, and A_{c_i} is the set of all possible assignments to the random variables represented by  v_{i_1}, v_{i_2}, ... v_{i_{|c_i|}}. In other words, each clique has associated with it a function from possible assignments to all nodes to the nonnegative real numbers.
I find this needlessly complex and somewhat confusing.  Here is how I would word it:
* a set \Phi of potential functions \phi_{c_i} : A_{c_i} \rightarrow R^+, where each c_i  is a clique in G, and A_{c_i} is the set of all possible assignments to the elements of c_i.  In other words, each clique has associated with it a function from assignments (to each element of the clique) to nonnegative real numbers.
It's not clear to me why the definition needs to mention both i's and v's, or why such schematic letters are even necessary for the definition at all.  (Can't we just identify the vertices of the graph with random variables rather than invoking some "representation" relation between them?)  However, I'm hesitant to just unilaterally edit the article, as I'm not very knowledgeable about these things.  Thoughts?  Dbtfz (talk - contribs) 02:21, 4 February 2006 (UTC)
replies: text: I went ahead and revised the passage mentioned above as well as much of the rest of the article.  Anyone who is interested, please review and revise as needed.  Dbtfz (talk - contribs) 06:02, 10 February 2006 (UTC)
replies: text: I am not convinced of the basis and need for the requirement of nonnegative real numbers. Is there a reason (ideally, with a reference) for such a restriction, or could the restriction not rather be omitted? See for example the antiferromagnetic Ising model (an Ising model being considered as a special case of a Markov network). --Chris Howard (talk) 16:46, 19:55, 4 May 2008 (UTC)
replies: text: Not sure, but I think its a notational problem: if we write f_k=\exp(\phi_k) then the f_k are non-negative, and the phi_k are merely real, positive or negative. Some texts seem to call the f_k the "potential functions", while others call the phi_k the potential functions; thus the source of confusion?! linas (talk) 18:21, 28 August 2008 (UTC)
replies: text: Thanks, linas, it´s fine now after this change you made. --Chris Howard (talk) 23:55, 30 December 2008 (UTC)
thread_title:  Potential functions 

message: The three Markov properties are not in general equivalent. Equivalence hold however for distributions, which are positive everywhere, see for instance Lauritzen, `Graphical models' or Koller, `Probabilistic Graphical models'.  —Preceding unsigned comment added by Derfugu (talk • contribs) 15:50, 17 November 2010 (UTC)
thread_title:  Equivalence of Markov properties 

message: A Markov network is similar to a Bayesian network in its representation of dependencies, but a Markov network can represent dependencies that a Bayesian network cannot, such as cyclic dependencies.
Shouldn't it be such as cyclic independencies ? Also, this sentence gives the impression that MRFs are a generalization of BNs, when BNs can actually represent independencies that a MRF cannot [1, p. 393 -- chapter available online]
[1]	C. M. Bishop. Pattern Recognition and Machine Learning. Springer, 2006.
-- 139.165.16.117 at 11:28, 17 July 2007
:I've fixed the latter problem. Took 15:00, 7 August 2007 (UTC)
replies: text: Hold up there; it's true that BN can represent conditional INdependencies that a Bayesian network cannot, but any Bayesian network can be converted into a Markov network representing the same dependencies (as well as new ones).  --12.17.136.181 00:40, 11 August 2007 (UTC)

text: A MRF can represent any joint PDF and thus represent any (in)dependencies provided you give the MRF the right clique structure, so BNs are a subset of MRFs. However, unlike an MRF, a BN is an explicit product of (directed and acyclic) conditional probabilities times a prior, so independence constraints can easily be “hardwired” into a BN (e.g. by using a factorised prior). It is generally more difficult to achieve (and to preserve under adaptive training) such constraints in an (undirected and cyclic) MRF.  — Preceding unsigned comment added by Teleprinter Sleuth (talk • contribs) 17:50, 17 November 2011 (UTC)
thread_title:  MRF vs. Bayesian Network 

message: Asterion85 (talk) 22:18, 4 May 2008 (UTC)
replies: text: I made a clarifying edit to the page; however, I reversed f and phi, since I think the more common usage is to use phi to denote the field, and use f for the distribution. linas (talk) 18:36, 28 August 2008 (UTC)
thread_title:  Log-linear model 

message: There are many undefined variables in this equation.
: P(X=x) = \frac{1}{Z} \exp \left( \sum_{k} w_k^{\top} \phi_k (x_{ \{ k \}}) \right) 
Could someone specify what 'P' and 'w_k' represent in this equation??  JEIhrig 18:46, 12 April 2009 (EST)
:P(X=x) means "the probability that the random variable X takes a certain value x (here X is a vector: all values in the network). But yes, what the heck is w, and what do the k indices loop over?  — Preceding unsigned comment added by 119.75.27.50 (talk) 07:53, 12 July 2013 (UTC)
thread_title:  undefined 

message: I am confused by this article. The term Markov network is not referenced, can a reference be supplied please? Other terms such as Markov random field are in widespread use. Also, a MRF is surely a probability function that satisfies one of the Markov properties. So then it is a THEOREM not a DEFINITION that it can be factorized in terms of cliques.
I propose to edit the article along the above lines. Objections and clarification invited? Have I got it wrong?
Megadeff (talk) 18:46, 29 May 2009 (UTC)
replies: text: The factorization definition is usually used as it is stronger than the Markov property definition: i.e. it is possible to construct a graph which satisfies the Markov properties but doesn't factorise (an example on a 4-cycle is given here). Under various conditions (positivity of the density or decomposability of the graph) the definitions are equivalent. —3mta3 (talk) 10:50, 30 May 2009 (UTC)

text: As for the name, personally I would call it an undirected graphical model.—3mta3 (talk) 11:13, 30 May 2009 (UTC)
replies: text: Thanks. Physicists call things satisfying the clique product formula a 'Gibbs state' or 'Gibbs random field'. Such use has been around a very long time. Maybe in statistics the product formula has taken over, and if that is what they want...fine. But could we keep the word 'Markov' to mean just that? Maybe a better solution is to develop the page on MRF, and let the statistical applications reign here.  Megadeff (talk) 08:41, 1 June 2009 (UTC)
replies: text: I think the two terms are related enough that it makes sense to keep them on the same article for now, though I would agree to renaming it to MRF, as that seems to be the more commonly used term. As for the definition, we could probably mention them both, my impression is that the differences only arise in pathological cases, so it seems like most people don't really differentiate between the two (well they don't appear to in the Kindermann and Snell book). —3mta3 (talk) 17:34, 1 June 2009 (UTC)

text: I've renamed and changed it to include both definitions, and tried to clarify the difference. —3mta3 (talk) 18:59, 2 June 2009 (UTC)

text: I am interested in the 4-cycle example you mentioned, where, if I understood you correctly, the 4-cycle is an independency map of the probability distribution (or else: what does it mean that the Markov property is satisfied?) but the distribution cannot be factorized using the 4 clique potentials. Skimming through the paper, the only image of a 4-cycle I found is on page 12. Is that the one you are referring to? -- 131.159.0.7 (talk) 19:30, 28 July 2009 (UTC)
replies: text: Yes, that was the example I was referring to, and what I meant by the Markov property being satisfied. I haven't actually read that paper, so can't really comment on anything else in it, but the same example also appears in this paper (Examples 4 and 7) where they explain why it doesn't factorise. —3mta3 (talk) 16:37, 29 July 2009 (UTC)

text: Okay, I've cleaned up the obvious errors. Personally, I wouldn't separate the articles, as they do refer to the same idea (a probability distribution with Markov properties over a graph), unless there is a large amount of physics specific material (in which case I think it would be better to keep the main article, and have a Markov random fields in physics type sub-article). —3mta3 (talk) 16:08, 27 July 2009 (UTC)
replies: text: I'll think it over. Megadeff (talk) 18:56, 27 July 2009 (UTC)
thread_title: Markov network: definition or theorem?

message: Shouldn't one say somewhere that the underlying graph has to be finite? When this is not true, many of the stated results do not hold in general (e.g., the equivalence between local and global Markov properties), because of the possibility of phase transitions. Such a remark would be particularly useful, since the first example given is the Ising model, which is very often defined on infinite graphs.--YVelenik 15:36, 7 April 2010 (UTC)
thread_title: Finite graphs

message: I took out the following two paragraphs from the definition because they are very badly worded and confusing and not really needed in a definition of a Markov random field. They were added by user 131.159.0.7.
:In other words, a set of random variables X is a Markov random field with respect to a graph G if the joint probability distribution P(X=x) over  if and only if graph separation in G implies conditional independence: If two nodes corresponding to X_i \in X and X_j \in X are separated in G after removing from G a set of nodes Z, then P(X=x) must assert that X_i and X_j are conditionally independent given the random variables corresponding to Z. If this condition is satisfied, we say that G is an independency map (or I-Map) of the probability distribution. 
:Many definitions go even further and require that G be a minimal I-Map, i.e. an I-Map from which none of the edges could be removed without destroying its I-Mapness. (It is reasonable to require this as it leads to the most compact representation, which involves as few dependencies as possible; note that a complete graph is trivially an I-Map.) In the case where G is not only an I-Map (i.e. it represents no independencies that are not indicated by P(X=x)) but also represents no dependencies that are not indicated by P(X=x), G is called a perfect map of P(X=x), as it represents precisely the set of independencies indicated by P(X=x).
However, perhaps the fact that the graph G with respect to which a random field is a Markov random field is sometimes referred to as an independency map or I-Map should be mentioned in the article? --Gustav (talk) 22:25, 1 January 2012 (UTC)
thread_title: Independency map

message: This article uses phi and f exactly backwards from how its commonly used; this is most irritating.  I once changed this around, but someone has changed it back.  I believe that phi = log f is the common notation, right? phi's are always potentials, and never probabilities. That is, 
:P(x)=\prod f = \exp \sum \phi
Can we please put it back? 
...Hmm. Its even worse that that. The current article uses f to stand for the delta function, and w for the potential. This seems like a doubly-non-standard notation, and goofy terminology to boot. Yeah?  Who talks like this? linas (talk) 14:17, 9 April 2012 (UTC)
replies: text: Computer scientists and econometricians talk like this. They have a rich tradition of making well-known maths look amazing and new by obfuscating notation. Sadly, when they do come up with something genuinely new and interesting, they have all of us scratching our heads... 2A01:CB0C:CD:D800:9047:E0B9:299:591D (talk) 08:28, 9 August 2022 (UTC)
thread_title:  backwards notation 

message: The introduction to the current article has the sentence: "When the probability distribution is positive, it is also referred to as a Gibbs random field, because, according to the Hammersley–Clifford theorem, it can then be represented by a Gibbs measure."   There is nothing preceeding this statement that explains what "the probability density" is.  There is no explanation of what a "positive" probability density is.  My guess is that a "positive" density is a density function that is strictly greater than zero on its domain. 
Tashiro (talk) 15:41, 11 October 2012 (UTC)
replies: text: This bothers me too. In the finite case (finite graph and finite target space for the pointwise values of the fields (states)), I suspect it means that each state has positive probability. I agree that this should be stated explicitly. But it's particularly important in the case when the graph is infinite, because then the state space is uncountable. Maybe then it means that each set of states defined by finitely many conditions (at finitely many points of the domain graph) should have positive measure. These are the open sets in the natural compact-open topology on the state space, if the target space is finite. 178.38.78.134 (talk) 02:51, 22 January 2015 (UTC)
thread_title: When the probability distribution is positive?

message: It would be nice to refer to "random field" and explain each word in the term. Compare for example with https://en.wikipedia.org/wiki/Gaussian_free_field. A random field does for example not need to be discrete. Here this is assumed to be the case without much thought. If the Markovian property is only defined for discrete fields, that must be stated as the reason for jumping to such a very specific model. I bet if you give a mathematician the mathematical objects of "random field" and "Markov property", he will come up with totally different things. :-) Anne van Rossum (talk) 13:49, 16 August 2014 (UTC)
thread_title:  Markov random field definition 

message: In the definition of Local Markov property -- should the neighborhoods be finite? Should each vertex of the graph have finite degree? 178.38.78.134 (talk) 02:41, 22 January 2015 (UTC)
thread_title:  Finite degree? 

message: The definition says "Given an undirected graph G = (V, E), a set of random variables ... form a Markov random field with respect to G  if they satisfy the local Markov properties:" and then lists three properties: the Pairwise Markov property, the Local Markov property and the Global Markov property. Are these three the "local Markov properties" that the first sentence of the Definition mentions? Then it is a little confusing that the "Global Markov property" is one of the "local Markov properties". Or does the phrase "local Markov properties" only refer to the second one of the three? 80.235.82.99 (talk) 14:17, 6 March 2015 (UTC)
replies: text: I agree, this is atrociously badly written. It seems to be symptomatic of the field. I am waiting for some actual mathematicians to take an interest and write a decent textbook about it. 2A01:CB0C:CD:D800:9047:E0B9:299:591D (talk) 08:26, 9 August 2022 (UTC)
thread_title:  Local Markov properties in Definition 

message: The article lead states that "in the domain of physics and probability, a Markov random field is a set of random variables having a Markov property described by an undirected graph". So how are Markov random fields relates to physics? —Kri (talk) 22:27, 24 June 2015 (UTC)
thread_title:  Physics? 

message: Still not very clear, though. Does this field attract people that are unusually unskilled in mathematical writing? (Computer scientists, econometricians...). 2A01:CB0C:CD:D800:9047:E0B9:299:591D (talk) 08:25, 9 August 2022 (UTC)
thread_title:  "is particularly clear in the following formulation..." 

message: The section Clique factorization begins as follows:
"As the Markov property of an arbitrary probability distribution can be difficult to establish ...." 
But he previous section, Definition, defines several kinds of "Markov property" only for random variables and not for distributions.
Or am I missing something?
In case I am not missing something, I hope someone knowledgeable about this topic can make this passage clearer. 
Also: The article uses the word "variable" when I am almost sure that it means "random variable".
If that is correct: That might be fine for mathematicians in this field talking among themselves. It is not fine for an encyclopedia. Spend the extra minuscule bit of energy and make Wikipedia clear to the reader. 2601:200:C000:1A0:1922:FB59:1D5E:AB37 (talk) 03:38, 26 August 2022 (UTC)
thread_title:  Unclear passage