Questions tagged [graphical-model]

Also called Probabilistic Graphical Model, used for statistical models expressed via graphs, causal or not. ("graph" as in graph theory). See e.g. <https://en.wikipedia.org/wiki/Graphical_model>

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How to exploit the similarity between multiple different entities when matching them against a target entity?

As a bit of background: I'm currently working on a record linkage problem, where I have different persons (entities) coming from different databases and I want to match them to a target entity in ...
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How to check if conditional independence holds from CPDs

These CPDs are presented in the Probabilistic Graphical Models course on Coursera as examples of conditional independence and conditional dependence, respectively. I have a vague idea of why the first ...
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Integrating a variable out of a distribution - The Graph Elimination Algorithm

Consider the graphical model below for 6 binary variables. It defines a joint distribution $\begin{align} p(x_1,x_2,x_3,x_4,x_5,x_6) &= p(x_1) p(x_2|x_1)p(x_3|x_1)p(x_4|x_2) p(x_5|x_3)p(x_6|x_2,...
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How does graphical model of a GP look like?

I'm trying to understand the difference between GP and Markov process. I couldn't find answers on the internet. I figured that graphical models can tell the difference, hence my question.
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Factorization of a completely connected undirected graph with pairwise compatibility functions

Given a completely connected undirected graph (V,E) such that $V=(x_1,\dots,x_5)$ and $E = ((x_i,x_j)_{i<j})$ for $i,j =1,\dots,5$, it is known that there exists a factorization $$ P(x_1,\dots,x_5) ...
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40 views

Energy function of Restricted Boltzmann Machine (RBM)

The energy function for RBM (Restricted Boltzmann Machine) is defined as $$ E(v,h) = -\sum_{i,j} w_{ij} \, v_i \, h_j -\sum_i a_i \, v_i - \sum_i b_i \, h_i $$ with the joint distribution $$ \tag{1} p(...
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Graphical Models Showing Independence Relations

I am attempting an old assignment question on graphical models. I am given a paragraph of information and asked to draw a directed graphical model showing the relationships between the variables and ...
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What does the graph for a convolutional layer look like?

I like to draw out computational graphs when I work on small assignments for computing gradients etc. and was wondering what the graph for a convolutional layer would look like? I can't seem to find ...
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23 views

Complete graphs have different v-structures?

From the book "Probabilistic Graphical Models", here says two complete graphs have different v-structures. As I understand, v-structure is like "X->Z<-Y" without edge between X and Y. If so, ...
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d-separation in Bayes Network vs separation in undirected graph

I've been teaching myself about d-separation and am trying to answer the following question. Given the graphs below, write down all conditional independence relationships involving the random variable ...
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The implementation of variable-to-factor and factor-to-variable messages?

I read this tutorial on the implementation of CRF and got to know that the normalization is the sum-product message passing. And I also know that there are two types of messages on factor graph: ...
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Why are exogenous variables not used in inference/recognition networks?

I have been working lately a lot on amortized variational inference. That is, doing variational inference using neural networks to approximate a variational distribution (such as in Kingma and Welling ...
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Learning Causal Graph from data

I am quite new to the theory of causal graphs, but from what I understand they are DAG, like Bayesian Networks. Since we have structure learning methods for Bayesian Networks like score based ...
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27 views

The difference between forward algorithm used in CRF and the variable elimination?

I found that in the forward algorithm used in the CRF(and perhaps also in the HMM) the mechanism applied is almost the same as that in the variable elimination(VE) except that the emission ...
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Computing Local Evidence for Bayesian Networks

I am reading through Kevin Murphy's "Machine Learning: A Probabilistic Perspective" book. I'm interested in understanding how to do exact bayesian inference over a tree structure, as discussed in ...
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75 views

How can I understand the complex regression models?

I can understand how it works when there are two variables in the linear regression model(the shaded circles represent the observed variables, and the white ones the latent variables): We can draw ...
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Action of Fisher information in “Latent Variable Model Selection…”

I'm having trouble understanding the role of the Fisher information matrix in the assumptions of Chandrasekaran et al. 2012. In the paper, the authors define the Fisher information matrix (i.e., ...
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applied papers on probabilistic generative models and inference engines

I am looking for applications papers where people choose some task on which they will do Bayesian inferencing and graphical modeling, and then build an inference engine to infer latent parameters. And ...
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35 views

Markov Blanket of two nodes?

I'm trying to solve a question and it has asked for (I feel like I'm confused by everything, would apprecaite some help, thanks), the Markov blanket of {c,d}. From what I've read so far, Markov ...
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69 views

How do we reuse and store the output of an MCMC?

For many Bayesian models, the posterior distribution is intractable... a solution is then to sample points from this unknow distribution with a Markov Chain Monte Carlo (MCMC). But at the end, how do ...
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Why are the additional set of parameters in discriminative models necessary(in Minka's 2005 paper)?

In a short paper titled Discriminative models, not discriminative training by Tom Minka, it says that the discriminative training might work better because it has two sets of independent parameters ...
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Do GGMs model partial correlation or conditional independence or both?

Post that says partial correlation != conditional dependence/independence: https://www.quora.com/Does-a-partial-correlation-of-0-imply-conditional-independence Post above points to following paper: ...
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Can Conditional Random Fields(CRFs) become a discriminative model by changing partition function?

In Introduction to conditional random fields, page 24, equation (2.18) and (2.19), the linear-chain CRFs is defined as: $$p(y|x) = \frac{1}{Z(x)}\prod_{t=1}^{T}exp\{\sum_{k=1}^{K}\theta_kf_k(y_t,y_{t-...
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58 views

What is the relationship between discriminative and generative and directed and undirected graphical models?

I feel that most generative models happen to be DGM(directed graphical model), and most discriminative models are UGM(undirected graphical model). Is there any correlation between these concepts? ...
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Why does Judea Pearl call his causal graphs Markovian?

In his texts on causality, Judea Pearl always refers to the simplest graphs he uses, i.e. the acyclic graphs with independent confounders, as Markovian. I don't see why these graphs contain anything ...
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Graphical model of phylogenetic linear regression

I have been trying to give the graphical model of the PGLS (phylogenetic generalized least squares) but can't seem to fully understand something. PGLS is a generalization of the general linear ...
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21 views

Is the Markov Network (Markov Random Field) property biconditional?

As far as I know, the property of a Markov Random Field is defined as follows: Let $G = (V, E)$ be a Markov Network. Let $X, Y, C \subseteq V$. If every path from a vertex in $X$ to a vertex in $Y$ ...
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Convert joint probability distribution in Gaussian graphical model to the general form in the general graphical model?

In the Gaussian graphical model, the joint distribution $p(x_1,\cdots,x_n)$ of p continuous random variables is a multivariate Gaussian: $$ p(x_1,\cdots,x_n)=\frac{1}{(2\pi)^{p/2}|W^{-1}|^{1/2}}exp(-\...
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113 views

What is the best method of visualising binary classification preditcions?

I have 2 columns in a dataset, "Device" and "Classification". I want to plot a graph that shows how well the "Classification" result is predicted given the data in the "Device" column. Both columns ...
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29 views

Level curves and functions of pair of Random Variables?

I came across the following question: I tried solving it, the following is my 1st attempt:(2nd method at the end) 𝑃[𝑊≤𝑤]=𝑃[𝑋𝑌≤𝑤]=𝑃[𝑌≤𝑤/𝑋] And then I simply double integrated keeping the ...
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144 views

How to determine if two directed probabilistic graphical models are I-equivalent?

I'm trying to figure out how to determine if these two models are I-equivalent. Google didn't properly give me a solid answer so far. Any idea on to determine it? Thank you.
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43 views

In directed acyclic graphs, is there a dependency in opposite directions?

Suppose we have this graph: (a) ==> (b) ==> (c) Does this mean that P(a|b)=P(a) because the arrows indicate that b is dependent on a and not the other way? If not, then why do we use arrows?
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Why is a single element getting better results than the sum?

I need an intuition how this is possible - Imagine I have a DAG(directed acyclic graph), so a graph without cycles and only directed edges; Now for every DAG with $d$ nodes there is (at least) one ...
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90 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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Can we apply Gaussian mixture model to all kinds of scensrios where some variables are unobserved?

I learned from this answer that: A mixture distribution combines different component distributions with weights that typically sum to one (or can be renormalized). A gaussian-mixture is the ...
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Graph problem with X nodes looking for Y paths with the most similar length. (shortest path / Chinese post man problem)

There is exactly 1 start node and 1 end node. There are also X (in this case 7) nodes, each connected to all other nodes and the start and end node with different lengths of the paths. The visitor ...
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74 views

What is the difference between homogeneous Markov chains and unhomogeneous Markov chains?

I learned that a Markov chain is a graph that describes how the state changes over time, and a homogeneous Markov chain is such a graph that its system dynamic doesn't change. Here the system dynamic ...
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29 views

Why does different factorisation matter in Markov networks?

I have been reading about Markov Networks that given some set of factors we can construct a unique graph G but not the other way around: "It should also be noticed that, given a set of factors, the ...
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69 views

Conditional Independence in Bayes Network

I have just started working through Michael Jordan's notes on Probabilistic Graphical Models and seem to be stuck on the exercise on page 5. I summarize the question here: Suppose $G = (V,E)$ is a ...
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43 views

Tool to draw neural network directed graphs [closed]

I would like to reproduce the kind of graphs used in the book Deep learning. Do you know the tool which can do that kind of graphs? Ideally I would like the apparence to be exactly the same. Here is ...
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29 views

forward sampling for Bayesian network with continuous variables and equation-based causal relationships

I have a physical system which can be represented by the following Bayesian network. It has the following characteristics 1) The encoded variables are continuous variables 2) The causal ...
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How do PGMs factor in to modern ML?

I just finished the three-part series of Probabilistic Graphical Models courses from Stanford over on Coursera. I got in to them because I realized there is a certain class of problem for which the ...
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130 views

Uncertainty and confidence in classification

I have a multi class classifier with 30-40 different classes. When i predict a class i would like to get an estimation about how certain my model is with its own prediction of that single sample . i....
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37 views

Understanding probabilistic inference graphs

I am having trouble understanding inference graphs. In the diagram below I understand the graph on the left (forward graph) where the arrows describe the direction that data flows when training for ...
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If I(G) = I(G') do they have the same skeleton and the same v-strucures except those in complete (sub-) graph?

I thought if two different graphs G, G' have the same skeleton and the same v-structure, then I(G)=I(G′) and I(G)≠∅. But does the converse also holds? In this case a complete graph doesn't apply ...
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Do all P-Maps for a distribution share the same skeleton or/and v-structures?

I learned that a distribution may have multiple P-Maps because even if we are sure about the independences in a graph we cannot gurrantee the causal direction, then I just wonder if those P-Maps share ...
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Resources for prerequisites to Probablistic Machine Learning Models

I am a self-learner and have done several machine learning courses but diving into Bayesian or Probabilistic Graphical Models I feel like my prior knowledge is inadequate. I have done some Probability ...
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51 views

How to use a G-Wishart distribution in stan

I would like to use the following kind of prior in a Stan simulation $$ f_{K \mid G} (k \mid g) = \frac{1}{I_g(b,D)}|k|^{\frac{b-2}{2}} \exp \biggl \{ -\frac{1}{2} \text{tr} (Dk) \biggr \}\mathbb{1}_{...
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132 views

EM Algorithm for Bayesian Networks with missing data

Setting: learning parameters of Bayesian Network (BN) with missing data. Algorithm: Expectation-Maximization. Question: suppose I am in the M-step, and that in the complete data there are no ...
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111 views

Is this formula related to the energy function / boltzmann distribution?

I am reading research done in this thesis. On page 20-21 (section 2.1) of the thesis, it describes the following stochastic model: ... let us focus for the rest of this chapter on time-varying ...