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{{Distinguish|Graph factorization}}
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In [[probability theory]] and its applications, a '''factor graph''' is a particular type of [[graphical model]], with applications in [[Bayesian inference]], that enables efficient computation of [[marginal distribution]]s through the [[sum-product algorithm]]. One of the important success stories of factor graphs and the [[sum-product algorithm]] is the [[code|decoding]] of capacity-approaching [[error-correcting code]]s, such as [[LDPC]] and [[turbo codes]].
 
A factor graph is an example of a [[hypergraph]], in that an ''arrow'' (i.e., a factor node) can connect more than one (normal) node.
 
When there are no free variables, the factor graph of a function ''f'' is equivalent to the [[constraint graph]] of ''f'', which is an instance to a [[constraint satisfaction problem]].
 
==Definition==
A factor graph is a [[bipartite graph]] representing the [[factorization]] of a function. Given a factorization of a function <math>g(X_1,X_2,\dots,X_n)</math>,
:<math>g(X_1,X_2,\dots,X_n) = \prod_{j=1}^m f_j(S_j),</math>
 
where <math> S_j \subseteq \{X_1,X_2,\dots,X_n\}</math>, the corresponding factor graph <math> G=(X,F,E)</math> consists of variable vertices
<math>X=\{X_1,X_2,\dots,X_n\}</math>, factor [[vertex (graph theory)|vertices]] <math>F=\{f_1,f_2,\dots,f_m\}</math>, and edges <math>E</math>. The edges depend on the factorization as follows: there is an undirected edge between factor vertex <math> f_j </math> and variable vertex <math> X_k </math> when <math> X_k \in S_j</math>. The function is tacitly assumed to be real-valued: <math>g(X_1,X_2,\dots,X_n) \in \Bbb{R} </math>.
 
Factor graphs can be combined with message passing algorithms to efficiently compute certain characteristics of the function <math>g(X_1,X_2,\dots,X_n)</math>, such as the [[marginal distribution]]s.
 
==Examples==
[[File:factorgraph.jpg|300px|right|thumb|An example factor graph]]
 
Consider a function that factorizes as follows:
:<math>g(X_1,X_2,X_3) = f_1(X_1)f_2(X_1,X_2)f_3(X_1,X_2)f_4(X_2,X_3)</math>,
 
with a corresponding factor graph shown on the right. Observe that the factor graph has a [[cycle (graph theory)|cycle]]. If we merge <math> f_2(X_1,X_2)f_3(X_1,X_2) </math> into a single factor, the resulting factor graph will be a [[tree (graph theory)|tree]]. This is an important distinction, as message passing algorithms are usually exact for trees, but only approximate for graphs with cycles.
 
==Message passing on factor graphs==
A popular message passing algorithm on factor graphs is the [[sum-product algorithm]], which efficiently computes all the marginals of the individual variables of the function. In particular, the marginal of variable <math> X_k </math> is defined as
:<math> g_k(X_k) = \sum_{X_{\bar{k}}} g(X_1,X_2,\dots,X_n)</math>
where the notation <math>X_{\bar{k}} </math> means that the summation goes over all the variables, ''except'' <math> X_k </math>. The messages of the sum-product algorithm are conceptually computed in the vertices and passed along the edges. A message from or to a variable vertex is always a [[Function (mathematics)|function]] of that particular variable. For instance, when a variable is binary, the messages
over the edges incident to the corresponding vertex can be represented as vectors of length 2: the first entry is the message evaluated in 0, the second entry is the message evaluated in 1. When a variable belongs to the field of [[real numbers]], messages can be arbitrary functions, and special care needs to be taken in their representation.
 
In practice, the sum-product algorithm is used for [[statistical inference]], whereby <math> g(X_1,X_2,\dots,X_n)</math> is a joint [[Probability distribution|distribution]] or a joint [[likelihood function]], and the factorization depends on the [[conditional independence|conditional independencies]] among the variables.
 
The [[Hammersley–Clifford theorem]] shows that other probabilistic models such as [[Markov network]]s and [[Bayesian network]]s can be represented as factor graphs; the latter representation is frequently used when performing inference over such networks using [[belief propagation]]. On the other hand, Bayesian networks are more naturally suited for [[generative model]]s, as they can directly represent the causalities of the model.
 
==See also==
* [[Belief propagation]]
* [[Bayesian inference]]
* [[Bayesian programming]]
* [[Conditional probability]]
* [[Markov network]]
* [[Bayesian network]]
* [[Hammersley–Clifford theorem]]
 
==External links==
* [http://www.volker-koch.com/diss/ A tutorial-style dissertation by Volker Koch]
* [http://www.robots.ox.ac.uk/~parg/mlrg/papers/factorgraphs.pdf An Introduction to Factor Graphs] by [[Hans-Andrea Loeliger]], ''[[IEEE Signal Processing Magazine]],'' January 2004, pp.&nbsp;28–41.
* [http://dimple.probprog.org/ dimple] an open-source tool for building and solving factor graphs in MATLAB.
* [http://people.binf.ku.dk/~thamelry/MLSB08/hal.pdf  An introduction to Factor Graph. Presentation from the ETH by Prof. Dr. Hans Loeliger]
{{No footnotes|date=September 2010}}
 
==References==
* {{Citation|contribution=Markov random fields in statistics|last=Clifford|year=1990|editor1-last=Grimmett|editor1-first=G.R.|editor2-last=Welsh|editor2-first=D.J.A.|title=Disorder in Physical Systems, J.M. Hammersley Festschrift|pages=19–32|publisher=[[Oxford University Press]]|url=http://www.statslab.cam.ac.uk/~grg/books/hammfest/3-pdc.ps}}
* {{Citation
    | last = Frey
    | first = Brendan J.
    | editor-last = jain
    | editor-first = Nitin
    | contribution = Extending Factor Graphs so as to Unify Directed and Undirected Graphical Models
    | title = UAI'03, Proceedings of the 19th Conference in Uncertainty in Artificial Intelligence, August 7–10, Acapulco, Mexico
    | year = 2003
    | pages = 257–264
    | publisher = Morgan Kaufmann }}
* {{Citation
    | last1 = Kschischang
    | first1 = Frank R.
    | authorlink1=Frank Kschischang
    | first2 = Brendan J. |last2=Frey |first3= Hans-Andrea |last3=Loeliger
    | title = Factor Graphs and the Sum-Product Algorithm
    | journal = IEEE Transactions on Information Theory
    | volume = 47
    | issue = 2
    | pages = 498–519
    | year = 2001
    | url = http://citeseer.ist.psu.edu/kschischang01factor.html
    | doi = 10.1109/18.910572
    | accessdate = 2008-02-06
    | postscript = . }}
* {{Citation
    | last = Wymeersch
    | first = Henk
    | title = Iterative Receiver Design
    | year = 2007
    | publisher = Cambridge University Press
    | url = http://www.cambridge.org/us/catalogue/catalogue.asp?isbn=9780521873154
    | isbn = 0-521-87315-0 }}
 
{{DEFAULTSORT:Factor Graph}}
[[Category:Graphical models]]
[[Category:Markov networks]]
[[Category:Application-specific graphs]]

Latest revision as of 01:14, 15 November 2014

Not maybe the very best-identified of Singapore points of interest, the Spice Backyard at Fort Canning Park is a veritable trove of Singapore's natural historical past, with over a hundred types of vegetation and bushes lining its trails.

The information contained on this materials is meant for common circulation solely. It should not be construed as a recommendation, or a suggestion (or solicitation of a proposal) to purchase or promote any financial products. The information offered doesn't take into consideration your specific funding goals, financial scenario or specific needs. Before you act on any advice which may be contained in this report, unbiased advice ought to be sought from a monetary adviser regarding the suitability of the investment product, taking into account your particular investment goals, monetary state of affairs or particular wants. All queries regarding the contents of this material are to be directed to Metropolis Index Asia Pte Ltd.

GPS Enbloc Sales Department has a devoted group of professional consultants focusing in collective gross sales and redevelopment of property growth. En bloc is exclusive in Singapore as authorities enhance the plot ratio, top restrictions, change of utilization zoning of a selected improvement – adjustments which enhance the worth of every house promoting collectively compared to selling individually within the open market. Full-time Gross sales Division I lease out my residential property and have to pay tax on my rental revenue and on the property. Am I being taxed twice? landed property in strata developments which are not accredited condominium developments underneath the Planning Act. c) Any non-residential, commercial or industrial property. Your Property May Be Featured Here!

Buying condos in Florida is very different from buying them in NYC. Right here, even today, most individuals purchase and sell condos for holidays and holidaying in cities like Miami and Orlando. South Florida is fast reworking into the most taking place part of the state. Condominium gross sales across Florida in 2009 had dipped and reached a bottom. But, from here on, the realty specialists predict that sales are more likely to surge, and it's the excellent time to acquire one if you want one in Florida.

Juwai.com has additionally referred purchasers with budgets of $200 million and A$320 million ($298.93 million) to actual estate agents overseas, based on Andrew Taylor, the property website's Hong Kong-primarily based co-chief government officer. Juwai.com's Taylor mentioned his portal has labored with customers as young as a 20-yr-previous pupil who was finding out within the United States and looking to purchase property there for his family again in China. This agreement also specifies whether or not one of the co-owners can modify the property. If yes, it additionally states what financial obligations this will incur on all of the house for sale in Singapore owners. Investor! Buyer! Let's Begin The Search Now ! An HDB flat purchased instantly from HDB; An HDB shophouse; and b) an individual who can present financial advantages to

Ultimately, owners at the moment are left with a choice of holding onto the property and paying considerably larger interest rates or selling their unit. Looking at how far more an proprietor must pay to keep up his unit, it's undoubtedly conceivable that those that had stretched themselves will contemplate promoting their models to minimise their monetary pressure. Naturally, the subsequent query is whether there are nonetheless property consumers in the market and what number of there are.

If you're taking out a housing mortgage from HDB chances are you'll borrow as much as ninety% of the acquisition price or the property's valuation depending on which is lower, topic to HDB's credit score evaluation. The quantity of housing loan that HDB can grant will rely on elements such because the buyers' age, month-to-month revenue and monetary standing. If you wish to get an HDB loan, you will have to first receive an HDB Mortgage Eligibility (HLE) letter before you decide to any buy. Chances are you'll apply for an HLE letter online by way of the HDB web site

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