Questions tagged [graphical-model]

Also called Probabilistic Graphical Model, used for statistical models expressed via graphs, causal or not. (Nb, "graph" as in graph theory, *not* as in figure or plot).

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Inference on a Gaussian random field / undirected graph?

Assume I have an undirected graph with $D$ nodes, and a $D$-by-$D$ matrix with edge strengths between $0$ (implying conditional indepedence given all other nodes), and $1$ (implying complete ...
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Question about using potential outcomes in DAGs in real world example

I am trying to understand how DAGs and potential outcomes look together. I came across these excellent posts (here and here, but I am trying to understand how this looks in a real world example. ...
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Posterior distribution is impossible depending on which prior hyperparameters are used?

Suppose we randomly select one of two coins and flip it. In that situation we have random variables $\alpha$ and $\delta$, where $\alpha$ tells us which coin we select, and $\delta$ tells us whether ...
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Conditional independence in EM algorithm

Let $X$, $\theta$ and $Z$ denote observed, parameter and latent nodes in a graphical model. The EM algorithm attempts to find a local maximum likelihood estimate $\theta^\ast$ for the likelihood of ...
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Parents in a directed acyclic graph vs a partial ancestral graph

In DAGs, parents are defined as follows: A is a parent of B if 'A -> B' edge is in the graph. In PAGs, there are mixed type of edges, so you can have A -> B, A o-> B. Obviously if A -> B,...
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Good example of a walk-through of the FCI algorithm to ensure all steps are done

The FCI algorithm is a common algorithm used for learning a Markov equivalence class of causal graphs from observational data. I am wondering if there are any good examples that walk through a causal ...
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Algorithm to check if there is an inducing path between two nodes - constructing maximal ancestral graph (MAG) given a DAG

In causal inference, one generally learns a Markov equivalence class of causal graphs when trying to reconstruct causal structure from data. This is known as a maximal ancestral graph (MAG). I am ...
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Why do I even need DeepWalk and Node2Vec when I can build a visual graph structure?

While studying DeepWalk, I started wondering why I need "DeepWalk" when I can build a graph from data and visualize the structure of a graph. With a visualized graph, I can see which nodes ...
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What is the most elegant way to express conditional independence on a line graph?

Consider a Markov graph $$x_1 -x_2-x_3-...-x_t$$ In such a graphical model, we have the conditional independence property $x_{s-1} \perp x_{s+1:t} | x_s \;\forall\; x=2,...,t-1$ and $x_{1:s-1} \perp ...
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Choice of Markov Random Field or Bayesian Network to model some causal and some non-causal links

Suppose you were modelling whether a person's ethnicity meant that they had different chances of getting a job due to discrimination. You have a couple of confounding variables e.g. deprivation - in ...
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Convergence of Gaussian random variables

Let $(f_n)$ be a sequence of 0-mean Gaussian densities on $\mathbb{R}^d$ and assume $f$ is limit of $(f_n)$. Question 1 How does one determine the type of convergence by looking at the corresponding ...
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State of the art methods for identifying DAG parameters

Say I have written down a directed acyclic graph (a causal model) with a few dozen variables. Moreover, I have a dataset with observations for many (though not all) of the variables. For simplicity, ...
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Doubts on a proof about graphical models

This is the third question I am asking about these notes http://www.stat.cmu.edu/~larry/=sml/DAGs.pdf .This time it is about the proof of a small theorem (page 426), that I report: Theorem: Let $G$ ...
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Is in a DAG every node an ancestor or a descendant?

This is the second question that I am asking here about these note about DAGs http://www.stat.cmu.edu/~larry/=sml/DAGs.pdf . When discussing the max-sum algorithm, they want to evaluate the marginal ...
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why is probability decomposition possible in Markov random fields?

I am reading the chapter about Graphical Models in Bishop's Pattern Recognition and Machine Learning, and in the book probability distribution of Markov random field is written as $$p(x)=\frac{1}{Z}\...
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How do Hidden Markov models classify sequential data?

How exactly do HMMs classify sequential data? I understand that this is a generative model, which models the joint probability distribution and provides us with the conditional probability of ...
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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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Directed graphical models and independence (exercise)

Context: this is Ex. 1 in these notes http://www.stat.cmu.edu/~larry/=sml/DAGs.pdf . The exercise asks to prove that, given a directed graphical model associated to a DAG (directed acyclic graph) $G$: ...
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Drawing graph of variance using R [closed]

I am a self -learner and try to learn statistics with R ,but i encounter with a problem i could not handle it such that I want to produce a graph of the variance of a binomial distribution with a ...
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Metropolis Hastings on hierarchical bayes update question:

[I have this somewhat complicated hierarchical bayesian model]1 Here the $y$ on $\theta$ are Poisson, $\theta$ are deterministically generated from the $att, def$ (and $home$). Then the last ...
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Proof that the Markov Blanket in a Bayesian Network is parents, children, and children's parents

I'm looking for a proof of this fact from wikipedia: The Markov boundary of a node $A$ in a Bayesian network is the set of nodes composed of $A$'s parents, $A$'s children, and $A$'s children's other ...
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Are all statistical models also causal models?

I'm just starting to learn about causal inference methods, focused on Pearl's do-calculus. So the point between Pearl's causal graphs and rules for manipulating causal graphs appears to be to turn a ...
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Reference Request: Variational Expectation-Maximization algorithm for Latent Dirichlet Allocation with an added time component

This link has a pretty good runthrough on the variational inference (via variational E-M) for LDA with calculations expanded and explained. I am now considering a modified LDA which adds a time ...
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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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Differentiating entropy in Reinforcement Learning as Probabilistic Inference

I am studying the paper Reinforcement Learning and Control as Probabilistic Inference: Tutorial and Review (https://arxiv.org/abs/1805.00909) and I do not understand how the author differentiate the ...
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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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Can we ignore graphs for inference in linear/Gaussian settings?

Assume I have a linear Bayesian network/graph like the following: where i can derive a joint pdf $$p(\mathbf{x})=p(x_1,x_2,x_3,x_4,x_5)=p(x_1)p(x_2|x_1)p(x_3|x_2)p(x_4|x_3)p(x_5|x_4)$$ Assuming that ...
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I-map Bayesian Network, Practical Explanation

I am having diffculty understanding the concept of an I-map in the context of Bayesian Networks. According to the PGM textbook by Koller & Friedman, an I-map is essentially a set of conditional ...
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Factor graph vs "moral graph" (preprocessing for belief propagation)

Some articles* about belief propagation formulate the algorithm in terms of undirected graphical models. Thus if we had a directed (acyclic) graphical model, it would need to be preprocessed into an ...
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Proving Equivalence between Multivariate distributions and Gaussian Bayesian Networks

I am studying Probabilistic Graphical Models by Daphne Koller. In Chap 7, the authors say the following. I can't convince myself of the highlighted part. Induction typically has a statement for n, ...
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Variational Autoencoders and Probabilistic Graphical Models

I am just getting started with the theory on variational autoencoders (VAE) in machine learning and I keep reading that VAEs belong to the category of Probabilistic Graphical Models (PGMs). As I ...
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Node2Vec: BFS versus DFS

In the paper that introduced Node2Vec, the authors mention the following: The neighborhoods sampled by BFS lead to embeddings that correspond closely to structural equivalence. Intuitively, we note ...
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2 votes
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modification of variable elimination for underflow

In Daphne Kollers book on Probabilistic graphical models exercise 9.3 asks the following Ex 9.3 Consider a modified variable elimination algorithm that is allowed to multiply all of the entries in a ...
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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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Stochastic Block's Model : Number of edges in a block?

So I am really confused about the number of (maximum possible) edges between two blocks in Stochastic Block's Model. In my understanding given two blocks or communities $b_r$ and $b_s$ containing $n_r$...
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What does the superscript triangle (△) symbol mean in graph/causality notation?

I am reading a paper called Robustly Disentangled Causal Mechanisms: Validating Deep Representations for Interventional Robustness and in stating a proposition the authors use a triangle superscript ...
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Which DAG would explain the lack of correlation between height and performance in NBA players?

A classic example of "selection bias" involves looking at the performance of professional basketball players. The example goes, among NBA players there is no correlation between height and ...
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Uniqueness of covariance graphical lasso

Denote the set of all $p\times p$ positive definite matrices by $\mathcal{M}^+$. For $\mathbf{\Sigma}\in\mathcal{M}^+$,define the following function : $$ u(\mathbf{\Sigma})=\log\det\mathbf{\Sigma}+\...
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2 votes
1 answer
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Bayesian and Markovian Networks: How do we obtain the probabilities at each node in a Bayesian or Markovian network

I just have a very basic 2 part question about Bayesian and Markovian networks. I suppose my confusion stems by trying to learn about these things through blog posts and videos, and not being able to ...
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EM Algorithm for Kalman Filters

Say I have the following dynamical system with unknown covariances for $w,v$. $$ z_n = Az_{n-1}+w\\ x_n = Bz_n +v $$ where $w \sim \mathcal{N}(0,Q)$ and $v \sim \mathcal{N}(0,R)$. I want to apply the ...
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Why do we need unary terms in Ising model (pairwise Markov random field)?

Ising model contains both $\phi(i,j)$ and $\phi(i)$. For example, consider a Markov random field with only two nodes $i$ and $j$, if $P=\phi(i,j) * \phi(i) * \phi(j)$, then we can also write $P=\phi(i,...
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Markov networks which are disconnected

I'm reading Kollers book on PGMs. Some of her examples to show the breaking down of theorems around independencies for non positive distributions involve either empty Markov networks or very sparse ...
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How are parameters in graphical models learnt?

This is a request of a good reference. I wanted to have a better understanding of graphical models and I am reading "Pattern recognition and machine learning" of Bishop. chap. 8 (Graphical ...
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Standard Error of ERGM Coefficients

I am trying to calculate the standard error of ERGM coefficients, which is estimated by MCMC sample. For an ERGM $P(y;\eta) = \exp[\eta^\top g(y) - \psi(\eta)]$, denote $\eta$ as the true parameter, $\...
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Training in Restricted boltzman machine

I am having doubt in training part of RBM's. I am confused between whats the difference between training RBM by block gibbs sampling and training RBM using contrastive divergence?
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Joint and Conditional Distributions in Bayesian Network

In such a graphical model, how can I express the conditional probabilities $p(x_4|x_1,x_2)$ and $p(x_4,x_5|x_1,x_2)$? My work: $p(x_4|x_1,x_2) = p(x_1,x_2,x_4) / p(x_1,x_2) p(x_1,x_2) = p(x_1)*p(x_2|...
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Intended selection bias

Sampling or selection bias is often presented as something that has to be overcome, avoided, or at least appropriately considered because it's a problem otherwise. I wonder how often situations arise (...
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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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How many parameters does the following Bayesian Network (Graphical model) contains?

These are the relations defined in the graph. P(D), P(E), P(A), P(C), P(G|D, E, A), P(I|E, A, C), P(T|E, C), P(F|E, A, C), P(S|A), P(J|G, I) Hello, I have this network that shows a graph (Graphical ...
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Interpretation of "log-likelihood" in hidden Markov model, and requisite computations involved

I am currently debugging a hand-coded implementation of a hidden Markov model, and as part of this, am scrutinising whether I have appropriately specified the log-likelihood computation algebraically. ...
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