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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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 ...
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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 $T$, ...
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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.
$$
p(\mathsf{x} \mid \theta) = \frac {1} {Z(\theta)} ...
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Why does Probabilistic Graphical Models by Koller claim that in Ising model when $w_{i,j}>0$, model prefers to align the spins?
In Probabilistic Graphical Models by Koller and Friedman, Box 4C introduces the concept of Ising Models -
Each atom is associated with a binary-valued random variable $X_i \;
> \epsilon \; \{+1,-1\...
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Intuition of conditional independence in DAGs
In the DAG above, we have $A$ conditionally independent of $E$ when $C$ and $B$ are observed (that is $A\perp E|B,C$), but not when only $C$ is observed (that is $A\not\perp E|C$). I don't have a ...
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dependence for variables that are not d-separated
I need to show that for a linear SEM having X->Y<-Z means that X and Z are dependent conditional on Y. For a linear SEM with errors that have finite variances this is doable, but for a model ...
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Calculating conditional distribution from an SEM
Below is an example from a set of slides:
Suppose the distribution(X, Y) was entailed by the SEM:
$$X \leftarrow N_X $$
$$Y \leftarrow 6X + N_Y$$
where $N_X, N_Y \sim Normal(0,1)$ and DAG $X \...
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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 ...
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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 ...
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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 &...
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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 ...
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Where to find proof of theorems in Probabilistic Graphical Models by Koller and Friedman?
I am self-studying Probabilistic Graphical Models by Koller and Friedman. Though the text is good, I am getting stuck with a few exercises, particularly the proofs of theorems that are given as ...
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Probabilistic Graphical Models, Daphne Koller, Exercise 3.22
Context:
I need some help regarding an exercise problem (3.22) from Probabilistic Graphical Models by Daphne Koller. This exercise is to explore cases where an algorithm for finding P-map fails when P-...
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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 ...
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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\...
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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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