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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Deriving conditional independence statements for causal graphs with selection nodes

In "basic" causal graphs / DAGs / probabilistic graphical models (PGMs), conditional independence statements can be derived using the d-separation criterion. How does this work if selection ...
Eike P.'s user avatar
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What is the meaning of graph singal for graph constructed from correlation matrix?

In the highly cited paper "The Emerging Field of Signal Processing on Graphs", the authors defined graph singal for a graph of N vertices as a vector of length N, with each element of the ...
Patrick's user avatar
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A question of "elementary imsets" in an ADMG

In [The m-connecting imset and factorization for ADMG models] (https://doi.org/10.48550/arXiv.2207.08963), it was mentioned the notation of an "elementary imset". The definition of ...
weihua li's user avatar
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Is i(G) equivalent to i_l(G), the local independency set of BN?

Reference: Koller, D., & Friedman, N. (2010). Probabilistic graphical models: Principles and techniques (Nachdr.). MIT Press. Bayesian network $\mathcal{G}$ encodes that for any node $X_i$, there ...
StorageGuard.Osaka's user avatar
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Converse of pairwise Markov property

Random vector $X$ follows a pairwise Markov property on graph $G=(V,E)$ if for any $(i,j) \notin E$, $X_i$ and $X_j$ are conditionally independent given $X_{V \setminus \{i,j\}}$. My question is, why ...
Phil's user avatar
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Causal discovery on partially known causal DAG

Often in data science, we have partial knowledge of the causal DAG structure. Regarding some of the possible edges in the DAG, we are in doubt. Are there any resources to tackle this setting? The ...
Anirban Chakraborty's user avatar
4 votes
2 answers
547 views

Isn't strong ignorability an incorrect assumption in complex causal structures?

I have seen that in many papers/competitions for causal inference, the assumption of strong ignorability is made - $P(Y^{x}\perp X\mid V)$, where $X$ is the treatment, $Y$ the outcome and $V$ ...
Anirban Chakraborty's user avatar
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Extremal function as defined in the paper Graphical Models for Extremes

The following is a question based on the paper Graphical Models for Extremes by Engelke and Hitz. For our purposes, exponent measure is a measure defined on $[0,\infty)^d - \{0\}$ with a density $\...
Phil's user avatar
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Factorization for P(A, B, C, D) that includes P(A, B | C, D) and its visualization

I was revisiting some basic concepts on graphical models and factorization of distributions and noticed that all the examples I see only have factors that include, at most, one conditioned random ...
echo66's user avatar
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How to write the likelihood for a multivariate gaussian linear model

I have a lasso-like bayesian graphical model where we try to estimate precision matrices between two conditions (0 and 1), $\Sigma_0^{-1}$ and $\Sigma_1^{-1}$, respectively. The model can be ...
Mangnier Loïc's user avatar
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Conditional independence statements for probabilistic motivation for linear regression

So the motivation for using the squared loss in linear regression can be written as the following (I think): Assume $\{(\mathbf{x}_i, y_i) \mid i = 1, \dots n\}$ are repeated independent samples from ...
Dylan Dijk's user avatar
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Graphical Lasso for estimating words network

I have a matrix whose columns are words and rows are different speeches by a person. Therefore, the i,j element of the matrix is the count of occurence of a word in a speech. I would like to estimate ...
unter_983's user avatar
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How to simplify the following conditional probability distributions using the given DAG?

Using the above DAG I need to simplify the following conditional probabilities: $$i) \quad p(x_4|x_1,x_2)$$ For this one I guess I can just remove the conditioning on $x_1$ (using the DAG) and the ...
Caporal Fourrier's user avatar
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Competitions/datasets fit for exploring Pearl's graphical causal models

Are there any competitions/challenges/datasets fit for testing Pearl's graphical causal inference methods? I do not necessarily mean live competitions. I would expect these setups to be different than ...
Anirban Chakraborty's user avatar
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1 answer
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How to understand the second rule of front door criterion?

In the Definition 3.4.1 of Pearl's causal inference book (Primer), the second rule for the front door criterion is "There is no backdoor path from $X$ to $Z$". But from my understanding, ...
bcxiao's user avatar
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Markov blanket - probability derivation

Is this correct reasoning? Let $x_i$ be a variable in a Bayesian Network and $\text{MB}(x_i)$ denotes its Markov blanket. Let us note that: $$ p(x_i \mid \text{MB}(x_i)) \propto p(x_i, \text{MB}(x_i))....
Elizabeth_Banks's user avatar
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Understanding the Ising Model and finding the MLE

In a binary pairwise MRF, the joint distribution is as follows: \begin{align} p(x\mid\theta) & = \exp\left(\sum_{s \in N} \theta_s x_s + \sum_{(s,t) \in E} \theta_{st} x_s x_t - \Phi(\theta)\right)...
dlu's user avatar
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Noise abduction for computing counterfactuals

Given observational data $X$ and knowledge of the true causal graph structure $\mathcal{G}$. How does abduction of the exogenous noise ($U$) for computing counterfactuals work? We don't have data ...
CausalQuestions's user avatar
2 votes
2 answers
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What variables need to be controlled for in this causal graphical model?

I have the below graphical causal model. I thought that when we apply the intervention i.e. do calculus we get to the graph on the right - that is deleting arrows going into the treatment (drug). to ...
Maths12's user avatar
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how to compute two bottom-up evaluations for discriminative Sum-Product Networks with Marginal Inference?

I am struggling with the two bottom-up evaluations for Sum-Product Networks with Discriminative Training with Marginal inference. In the paper "Discriminative Learning of Sum-Product Networks&...
Utkarsh Kathuria's user avatar
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Justification of definition 2.7.1 (Potential cause) in Causality by Pearl

In Causality - Models, Reasoning And Inference by Pearl, definition 2.7.1 says - Potential Cause definition: A variable $X$ has a potential causal influence on another variable $Y$ (that is ...
Anirban Chakraborty's user avatar
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Does a PAG (partial ancestral graph) have almost directed cycles with circular endpoints?

In https://www.jmlr.org/papers/volume9/zhang08a/zhang08a.pdf, a Maximal Ancestral Graph (MAG) is defined as: ...
ajl123's user avatar
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In what sense is one latent causal structure "preferred to" another? Definition 2.3.3 from Causality by Pearl

In Causality - Models, Reasoning, And Inference by Pearl, definition 2.3.3 reads as follows - One latent structure $L$ = $\langle D,O \rangle$ is preferred to another $L^{'}$ = $\langle D^{'},O \...
Anirban Chakraborty's user avatar
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What type of probabilistic assumption do we make in a MRF if we break a clique by pairwise potentials?

If we have a fully connected MRF with three random variables $a, b, c$, what probablistic assumption would we make if we break the joint potential of the three by pairwise potentials? $$ \phi(a,b,c) = ...
Adrian's user avatar
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Graph based variational Autoencoder with variable latent size

I'm trying to build a graph-based Variational-Autoencoder, which should be able to generate graph structures (adjacency matrices). So far, all the papers and models I've seen use a fixed latent vector ...
user3748950's user avatar
1 vote
2 answers
109 views

Derivation of the formula of the variance of a linear gaussian

I'm trying to understand this formula for the variance in Kohler's PGM text. It's theorem 7.3: Let $Y$ be a linear Gaussian of parents $X_1 .. X_k$: $$p(Y|X) = \mathcal{N}(\beta_0 + \beta^T\mathbf{x};...
Estimate the estimators's user avatar
3 votes
1 answer
151 views

What is an inducing path in causal inference?

In causal graphical models, an inducing path is defined as: [Definition Inducing Path] An inducing path relative to L is a path on which every non-endpoint node ...
ajl123's user avatar
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What is the suitable Graph Machine Learning model for dynamic graph forecasting with changing nodes and edges features

I have a graph where each node carries a feature value(s). The edges in the graph are weighted and each edge carries a single value (weight). The weight of the edge is some value calculated using the ...
Traveling Salesman's user avatar
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2 answers
311 views

Model causality: graphical models and PCA

If we build a graphical model (DAG) we (may) interpret the arrows as causal dependences. If we build a graphical model based on the variables returned by principal component analysis (PCA) we should ...
Thomas's user avatar
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Specifying a terminal node in a Bayesian Network

I am writing R script using bnlearn package for learning a DAG from a dataset. Is it possible to define a terminal node beforehand? ie I want a node Y to be the final one in the graph, which will not ...
mattara's user avatar
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7 votes
2 answers
121 views

What is a "directed path" in context of causal graphs?

I am going through Causal Inference In Statistics by Pearl and I have come across the definition of path and directed path (section 1.4, page 25). Path: A path between two nodes $X$ and $Y$ is a ...
Anirban Chakraborty's user avatar
1 vote
1 answer
67 views

Pearl's Causal Inference In Statistics, equation 3.11 - Calculation of group specific causal effects

In the book Causal Inference In Statistics by Pearl, page 63, while referring to the below DAG, it says - Thus to compute the $w$-specific causal effect, written $P(y|do(x),w)$, we adjust for $T$, ...
Anirban Chakraborty's user avatar
2 votes
1 answer
380 views

Why is computing the partition function expensive?

The joint distribution of a undirected graph can be factorized as a product of potential functions over the maximal cliques of an undirected graph. $$ p(\mathsf{x} \mid \theta) = \frac {1} {Z(\theta)} ...
GaryTheBaddy's user avatar
1 vote
1 answer
594 views

Intuition of conditional independence in DAGs

In the DAG above, we have $A$ conditionally independent of $E$ when $C$ and $B$ are observed (that is $A\perp E|B,C$), but not when only $C$ is observed (that is $A\not\perp E|C$). I don't have a ...
statzoo's user avatar
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Calculating conditional distribution from an SEM

Below is an example from a set of slides: Suppose the distribution(X, Y) was entailed by the SEM: $$X \leftarrow N_X $$ $$Y \leftarrow 6X + N_Y$$ where $N_X, N_Y \sim Normal(0,1)$ and DAG $X \...
ZWZWZW's user avatar
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3 votes
1 answer
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Layman's explanation for Finest Fully Randomized Causally Interpretable Structure Tree Graph (FFRCISTG) and NPSEM-IE

I am reading Single World Intervention Graphs (SWIGs): A Unification of the Counterfactual and Graphical Approaches to Causality, and they describe both Finest Fully Randomized Causally Interpretable ...
jacqui_suis's user avatar
5 votes
1 answer
307 views

Is it a confounder on not?

I have a following picture and the assumption that I can estimate the effect of Treatment on Growth by accounting for dT. However, I'm not sure if Unobserved confounder is actually a confounder - it ...
Maria Li's user avatar
2 votes
1 answer
43 views

Meaning of $\uparrow$ in below d-separation algorithm from Koller

In Probabilistic Graphical Models by Koller and Friedman there is an algorithm to find the nodes reachable from node $X$ via trails that are active, given conditioning set $Z$. What is the meaning of &...
Anirban Chakraborty's user avatar
2 votes
1 answer
75 views

Do Bayesian Network perfect maps need to be chordal?

In Probabilistic Graphical Models by Koller and Friedman, there is a proposition - The PDAG $\mathcal K$ returned by Build-PDAG is necessarily chordal. Build-PDAG is an algorithm that builds the ...
Anirban Chakraborty's user avatar
1 vote
0 answers
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How can I reduce data in a significantly smaller size without losing it's "Representational" significance? [closed]

So here I am trying to make sense of some test results from temperature chamber for testing electronics. Temperature vs Time Graph shows the final output of the test. The data recorded by temperature ...
Sajeev Pillai's user avatar
2 votes
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30 views

Choice of approximate posterior in variational inference with positive support

I have a simple probabilistic graphical model: $z \longrightarrow x$ where $z_i \sim Exp\left(\lambda_i\right)$ where subscript $i$ denotes the $i$th dimension and $x|z \sim \mathcal{N}\left(f\left(z\...
isle_of_gods's user avatar
1 vote
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22 views

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, ...
Douw Marx's user avatar
3 votes
2 answers
240 views

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. ...
Anirban Chakraborty's user avatar
4 votes
1 answer
192 views

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 ...
ajl123's user avatar
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2 votes
1 answer
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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 ...
user3269's user avatar
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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 ...
João Areias's user avatar
1 vote
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60 views

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 $I(X,W\cup Z,Y)$ $\implies$ $...
Anirban Chakraborty's user avatar
1 vote
0 answers
67 views

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,.....
user's user avatar
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1 answer
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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,.....
user's user avatar
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1 vote
1 answer
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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 ...
user's user avatar
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