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).

Filter by
Sorted by
Tagged with
0 votes
0 answers
4 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, ...
3 votes
2 answers
49 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. ...
3 votes
1 answer
25 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 ...
  • 185
1 vote
1 answer
16 views

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 ...
  • 4,802
2 votes
2 answers
55 views

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 ...
1 vote
0 answers
19 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 $I(X,W\cup Z,Y)$ $\implies$ $...
1 vote
0 answers
28 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,.....
  • 71
1 vote
1 answer
33 views

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,.....
  • 71
0 votes
0 answers
23 views

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 ...
1 vote
1 answer
36 views

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 ...
  • 71
3 votes
1 answer
43 views

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 ...
  • 71
2 votes
1 answer
35 views

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 ...
  • 71
1 vote
1 answer
46 views

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 ...
  • 71
1 vote
1 answer
24 views

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 ...
  • 71
1 vote
1 answer
38 views

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 ...
  • 71
2 votes
1 answer
33 views

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 ...
  • 71
1 vote
1 answer
49 views

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 ...
  • 71
0 votes
1 answer
37 views

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, ...
  • 71
2 votes
1 answer
31 views

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 ...
  • 444
1 vote
1 answer
22 views

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 ...
2 votes
1 answer
29 views

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 ...
  • 41
1 vote
1 answer
17 views

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 ...
  • 444
0 votes
0 answers
5 views

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 ...
0 votes
0 answers
24 views

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 ...
  • 1,242
0 votes
0 answers
12 views

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 ...
  • 473
2 votes
0 answers
68 views

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. ...
  • 29
7 votes
2 answers
390 views

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 ...
  • 288
0 votes
1 answer
35 views

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 ...
  • 211
1 vote
0 answers
22 views

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,...
  • 185
1 vote
0 answers
28 views

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 ...
  • 185
2 votes
0 answers
47 views

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 ...
  • 185
0 votes
1 answer
115 views

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 ...
  • 2,645
1 vote
0 answers
16 views

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 ...
  • 473
1 vote
0 answers
34 views

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 ...
  • 637
0 votes
0 answers
42 views

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 ...
  • 121
0 votes
0 answers
29 views

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, ...
  • 1,484
1 vote
2 answers
84 views

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 $G$ ...
  • 703
1 vote
0 answers
80 views

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 ...
  • 703
0 votes
0 answers
17 views

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}\...
1 vote
0 answers
36 views

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 ...
1 vote
0 answers
30 views

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-...
  • 11
1 vote
0 answers
28 views

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) $G$: ...
  • 703
0 votes
1 answer
237 views

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 ...
0 votes
0 answers
41 views

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 ...
  • 1
1 vote
0 answers
123 views

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 ...
  • 111
6 votes
3 answers
432 views

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 ...
0 votes
0 answers
33 views

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 ...
  • 1
0 votes
1 answer
33 views

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 ...
0 votes
0 answers
22 views

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 ...
0 votes
1 answer
99 views

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 ...

1
2 3 4 5
11