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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Implications of violating Bayesian network independence assumptions during inference
Consider the example Bayesian network below where $X \perp \!\!\! \perp Y $ (X is independent of Y).
Assuming that this is the true independence structure of the process that is generating the data, ...
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Problems with zero probability events in Bayesian Networks
In the book Probabilistic Graphical Models: Principles And Techniques by Daphne Koller, the author at one place (Box 3c), states the challenges in picking probabilities for a Bayesian network model. ...
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sigma-separation question in cyclic causal graph - understanding sigma-separation
Main Question
In https://arxiv.org/pdf/1807.03024.pdf, a generalization of d-separation in DAGs is introduced, called $\sigma$-separation for cyclic graphs.
I am wondering how $v_1 \perp v_6$ using ...
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ignorable assignment mechanism in causal studies
In the causal studies, there is so-called ignorable assignment mechanism. For instance,
The vast majority of causal studies assume certain versions of an
ignorable assignment mechanism, where the ...
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How many parameters on a Bayesian network
I'm taking Coursera's course on probabilistic graphical models, and I'm stuck on a question. The discussion forums there are dead, and I can't find any resource to help me, so I hope someone could ...
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Show that intersection property - $I(X,Y\cup Z, W)$ and $I(X,W\cup Z,Y)$ $\implies$ $I(X,Z,Y\cup W)$ - requires strict positive distribution
Question
It is stated in Probabilistic Reasoning In Intelligent Systems by Judea Pearl that, intersection property of information relevance axioms -
$I(X,Y\cup Z, W)$ and $I(X,W\cup Z,Y)$ $\implies$ $...
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Observed hidden variables in HMM
I am studying Hidden Markov Models and I'm trying to understand the following exercise:
Consider Hidden Markov Model with hidden states $h_{1:T} = \{h_1,...,h_T\}$ and observed states $v_{1:T}=\{v_1,.....
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Independence in Graphical model of $p(h_{1:T}|v_{1:T})$ of an HMM
I am studying Hidden Markov Models and I'm trying to understand the following exercise:
Consider Hidden Markov Model with hidden states $h_{1:T} = \{h_1,...,h_T\}$ and observed states $v_{1:T}=\{v_1,.....
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Compare Probabilistic Graphical Models and Probabilistic Reasoning In Intelligent Systems
I am new to probabilistic graphical models. I am currently reading Probabilistic Reasoning In Intelligent Systems by Judea Pearl. I found the first two chapters (and first half of third chapter, till ...
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Deriving the expression for $p(\mathcal{K})$ where $\mathcal{K} = \{(\mathbf{s}^k,\mathbf{d}^k), k = 1,..., K\}$
This is a follow up from this question.
Consider a model of diseases and symptoms. $s_i\in\{0,1\}$ is a binary random variable indicating whether the patient is showing the $i$-th symptom and $d_j\in ...
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Expression for $p(\mathbf{s}|\mathbf{d})$ and the respective Markov Network
I would like to check if I correctly derived my the expression for $p(\mathbf{s}|\mathbf{d})$. Here's the question:
Consider a model of diseases and symptoms. $s_i\in\{0,1\}$ is a binary random ...
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Quick way to determine the different independence assumptions
This question is different than my previous question in that I'm asking sort of a "meta" question.
Here's two graphical models (a Belief Network and a Markov Network):
I would like to ...
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Determining unconditional independence in Markov Networks
I would like to know whether $E \perp\kern-5pt\perp A $ in the following Markov Network and would like to know if my reasoning is correct:
So, since this is a Pairwise Markov Network, it factorizes ...
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Checking for conditional independence in graphical models
I would like to know whether $B \perp\kern-5pt\perp C | D,A $ and $D \perp\kern-5pt\perp A | B,C $ in the following two graphical models and would like to know if my reasoning is correct:
For the ...
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Is $C \perp\kern-5pt\perp D | A $ for the two graphical models? [duplicate]
I would like to know whether $C \perp\kern-5pt\perp D | A $ in the following two graphical models and would like to know if my reasoning is correct:
For the left model (Belief Network), here's my ...
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Is $B \perp\kern-5pt\perp C | A $ for the two graphical models?
I would like to know whether $B \perp\kern-5pt\perp C | A $ in the following two graphical models and would like to know if my reasoning is correct:
For the left graphical model, which is a Belief ...
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What exactly would be a perfect map in this situation? Is a perfect map a distribution which has the same independence assumptions?
I am currently studying Bayesian Reasoning and Machine Learning by David Barber, the 4th chapter exercise 4.7 (p 80). The exercise is the following:
Consider the following belief network:
Write down ...
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Is it always possible to find a joint distribution $p(x_1,x_2,x_3,x_4)$ consistent with these local conditional distributions?
I am currently studying Bayesian Reasoning and Machine Learning by David Barber, the 4th chapter exercise 4.1 (p 79). The exercise is the following:
Exercise 4.1
Consider the pairwise Markov network,
...
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How can I find Conditional Probabilties from dataset points of features (random variables)?
I am trying to solve the following at work and will dummify for the sake of making it easier to explain myself and getting an answer. My main query is about Step 4 below. But if something is wrong or ...
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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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Bayes graph model parameter dependencies problem with prior density
Suppose I have a graph model
with underlying density (denote $f(\theta_1|\theta_0):=f(\theta_1)$)
$$f(\theta|x)\propto \prod_{i=1}^k f(x_i|\theta_{1:i})f(\theta_i|\theta_{1:i-1}). $$
Suppose markov ...
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How can I model, and train a Bayesian Netwok from a given dataset for predicting cause-effect scenario?
I have a dataset that I used to train my ML model for prediction. I created my prediction models using XGBoost and Neural Networks.
Now I want to create another model that can give me the causal ...
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Why do we calculate probability distribution on leaf nodes at Sum-Product Networks (SPN)?
I am new to Sum-Product Networks (SPN) and still trying to understand some concepts.
I understand that we need to give inputs at the leaf nodes of SPN. But why do we have gaussian distribution and ...
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Are Graphical Models more accurate than simpler regression models and why?
I am looking at graphical models again--such as Bayesian networks and also undirected Markov Random fields. I was hoping to benchmark these models against simpler regression models, but was just ...
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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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