# 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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### Purpose of the hidden variables in a Restricted Boltzmann Machine

From the part titled Introducing Latent Variables under subsection 2.2 in this tutorial: Introducing Latent Variables. Suppose we want to model an $m$-dimensional unknown probability distribution $q$ ...
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### How to incorporate multiple likelihoods in a probabilistic graphical model with Stan?

Data composition: In beta testing of a video game, users were assigned tasks in a many-to-many relationship. At the end of every day, users were asked to self evaluate (for each task) whether they ...
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### Finding the regression coefficients that will remain invariant when an additional variable is added as a regressor

This is Problem 3.8.1(f) in Causal Inference in Statistics: A Primer, by Pearl, Glymour, and Jewell. The above figure implies that certain regression coefficients will remain invariant when an ...
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### Effect of treatment on the treated using the graph: Problem 4.3.2(b) in Causal Inference in Statistics: A Primer

From the above diagram, I need to find the effect of education on those students whose salary is $Y=1$. I was given the hint to use $E[Y_1 − Y_0|Y = 1]$. My attempt: I tried to expand the above ...
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### If the sample size is > 100, which graphical summarization is the best? [closed]

Box and whisker plot 3D doughnut plot Heatmap depicting expression levels Line plot Column scatterplot
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### Interpretation of concentration of posteriors in the limit of infinitely many independent versus dependent random variables

Disclaimer: the setup and specific example may not be a minimal example to illustrate the point, but I am not well-versed in these topics enough to construct a smaller example without accidentally ...
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### Why can't Bayesian and Markov networks represent all conditional independencies in a joint distribution?

From here: An I-map is said to be perfect if $I(G)=I(p)$. Given a distribution $p$, it is not always possible to find a DAG $G$ such that $I(G)=I(p)$. Consider a joint distribution over four random ...
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### Finding the probability based on the given graph

I have a graph $G_1\rightarrow G_2\rightarrow G_3\rightarrow G_4$ which has random variables and it's probabilities are given by $$P(G_i=1|G_{i-1}=1)=a$$ $$P(G_i=1|G_{i-1}=0)=b$$ $$P(G_1=1)=a_0$$ I ...
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### Predict edge weights from known connectivity but limited edge weight data

If I have a known adjacency matrix for a connected, undirected graph, and if I know some of the edge weights, what methods or techniques could I use to predict the weights of other edges? So in my ...
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### What is the name of this weakened naive Bayes assumption?

The naive Bayes assumption is just the assumption that a probability distribution over several variables factors into the product of all the one-dimensional distributions. This is equivalent to saying ...
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### Do we assume graphical LASSO explanatory variables to be normally distributed? And what if this assumption fails?

I am working on a graphical LASSO (GLASSO) shrinkage of the variance-covariance matrix of financial log-returns data for 10 years. I tested for normality and the Jarque-Bera test (but also other tests)...
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### Spirtes' example of d-separation not leading to independence in a directed cyclic graph with non-linear structural equations

In Spirtes (1995) there is an example (Fig. 4 on page 495, reproduced below) of a directed cyclic graph with non-linear structural equations in which $d$-separation of $X$ and $Y$ given $\{Z, W\}$ ...
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### imputing high percentage of missing data in multivariate time series

In a dataset with time-series, that is dependant on a given input, which the time-series are given only on an irregular cycle of 10-12 time steps that makes lots of missing observations what is the ...
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### Graphic model factorizing, marginalization

This is actually a probability marginalization question that I encountered in graphic models section of PRML by Bishop (question about equation 8.26 page 391). Assume I have the following graphic ...
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### Packages for autoregressive HMM?

I have data I'd like to fit a generalized HMM on: my observations $\{Y_t\}_{t=1}^N$ and my states $\{X_t\}_{t=1}^N$ are both time series. The specific task I'd like to do is decoding the states given ...
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### Do variational approximations capture the flow of influence or “conditional independence” relationships in graphical models?

Probabilistic Graphical Models (PGMs) are used to model all sorts of complex decision processes, such as medical diagnoses or robot positions, etc. In common machine learning textbooks, like ...
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### properties of Gaussian graphical model of sub-covariance matrix

I'm working on Gaussian graphical models. The precision matrix is $$\Theta = \Sigma^{-1}$$ What properties does the sub-matrix have? To be more specific, $$\Theta' = (\Sigma_{A,A})^{-1}$$ $A$ is the ...
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### What is the benefit of adding control flow to probabilistic programming?

I was watching an interesting video on the Pyro package in Pytorch for probabilistic programming. One of the things that they ...
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### Are nodes outside the markov blanket unconditionally independent?

Apologies if my question is deeply flawed, I've been working through a lot of material in the past few weeks and have a few blind spots here and there. On one level my question is this - given a ...
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### Strong ignorability: confusion on the relationship between outcomes and treatment

In the research area of potential outcomes and individual treatment effect (ITE) estimation, a common assumption called ''strong ignorability'' is often made. Given a graphical model with the ...
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### Deriving Hyperparameter updates in Online Interactive Collaborative Filtering

I've been going through "Online Interactive Collaborative Filtering Using Multi-Armed Bandit with Dependent Arms" by Wang et al. and am unable to understand how the update equations for the ...
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### Bayes network, what is the purpose of it?

I really don't understand what is the purpose of the Bayes network, actually how to implement it into a useful application. It all starts with the data. Let's assume I observe some universe and I ...
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### d-separation and implication of its theorem

Let $(X\perp Y | Z)_P$ represents the conditional independence of X and Y separated by Z. I am very confused about the following theorem about d-separation from Judea Pearl's text which says the ...
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### Causality: Implication of d-separation

I am confused about the following theorem about d-separations from Judea Pearl's causality textbook which reads as follow: "If sets X and Y are d-separated by Z in a DAG G, then X is independent of Y ...
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### Relationship between variational inference and sampling in a Boltmzann-machine-like network

In this paper concerning a Boltzmann-machine-like network and its variational mean field approximation, the authors write In the stochastic system as well as the deterministic system, units evolve ...
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### How to implement probabilistic PCA with missing data?

There is this site where they show you how to implement probabilistic PCA and they even mention that PPCA can handle missing data, however they don’t show how to tune the model. Do you know of any ...
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### The Graph Neural Network- understanding the back propagation mechanism

I am having trouble understanding the derivation of the back propagation algorithm for graph neural networks, as derived in Scarselli 2009 "The Graph Neural Network Model." (IEEE ...
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### Understanding deterministic models via probabilistic graphical model

I have read a few tutorials how we can think of deterministic neural networks with the help of probabilistic graphical models. Very often they would offer an image as seen bellow and say, our model is ...
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### Causality: Models, Reasoning, and Inference: Diagram Question

I am self-studying Causality: Models, Reasoning, and Inference, by Judea Pearl, and there is a question I am particularly stumped on. It reads like this: Problem Statement: Given this fragment of a ...
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### How do Graphical Models work in practice?

I know how graphical models work at a high level. I have knowledge about graphs in general, but the message passing is hard to understand and implement. I want to be able to understand what is going ...
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### Equivalence between directed and undirected graph?

I am confused over something that may have an obvious explanation I am missing. In Koller's Probablistic Graphical models textbook, page 945, it is said that a Markov network $A-B-C$ is equivalent ...
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### What is the difference between probabilistic graphical model and graphical model?

In this note: Lecture 1: Introduction to Graphical Models, it reads that Next, we will elaborate on the difference between Probabilistic Graphical Models (PGM) and Graphical Models (GM). In ...
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### graph convolution network

I am trying to understand papers and lectures on graph convolution networks but whenever I open some paper, I get lost on the very first page. I started with some videos like this and this and papers ...
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### Gaussian graphical models in regression

I am studying about the Gaussian graphical model (GGM). I have a $N\times D$ matrix X of my observations. The structure of the network has been found by using the graphical lasso method. It means I ...
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### Does it make sense to convert a time plot to a distribution

I'm plotting time series data where the 5th-95th percentile values overlap so it looks pretty messy. The data is based on a simulation which tracks military 96,000 military personnel purchase of ...
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### What is the relationship of Probabilistic Graphical Model and Bayesian statistics?

I am reading about Probabilistic Graphical Model on a machine learning textbook written by a computer scientist. Even though Probabilistic Graphical Model is more of an engineering approach, its idea ...
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### graph classification using ordered nodes and channel-like edges

I am running into a theoretical question and I'm not sure if what I'm hoping for is possible with the current technology. In short I am attempting to perform a graph classification. In one regard, my ...
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### Causal networks with correlated variables

I've been reading about Bayesian networks and one of the central assumptions these networks require is conditional independence. However, the problems I'm working on involve variables that are often ...
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### Which method would work the best to fill a matrix with empty cells based on the non-empty cells?

Say that you have s students and c courses and therefore a matrix of s x c grades. Suppose also that not all the students have coursed all the courses so there are empty cells in the matrix. Which ...
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### Why are autoregressive models neither directed or undirected, as described in the NADE paper?

In the paper Neural Autoregressive Distribution Estimation (Uria et al., 2016), NADE (and other autoregressive models) seem to be described as neither directed or undirected models: We’ve described ...
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### What is the intuitive meaning of P(A,B|C)?

I have come across graphical models where we have a node C that is dependent on both A and B. For example a node C with incoming arrows from A and B. In this case I know the intuition behind writing ...
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### In a graphical model of C dependant on both A and B, is it possible to get the joint distribution of C and B?

Imagine we have a graphical model of random variable C that is dependent both on A and B. This is like a node C which has two incoming arrows. One from node A and the other from B. An example is me ...
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### Understanding the graphical model for a GP for regression, from GPML (Rasmussen and Williams, 2006)

The book Gaussian Processes for Machine Learning (GPML) by Rasmussen and Williams (2006) provides a graphical model for GP regression but does not explain it in great detail, so I have a few questions ...
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### Identifying identical graphs or adjacency matrices of graphs

I was wondering if someone has a good idea for checking whether two graphs are the same (for example, based on an adjacency matrix). Ideally, in a computational efficient manner that can be done on ...
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### Problems with using Gibbs Sampling for Bayesian DAGs

Assume we want to sample from the variables of Bayesian belief network, which is a Directed Acyclic Graph (DAG), where we observe some of the variables, and do not observe the others. We can usually ...
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### Proof of a causual (line) Bayesian graph model

Given a simple Bayesian graph model, and $A$ is observed. A <---- B <---- C The joint model is $$p(A,B,C) = p(A\mid B,C)p(B\mid C)p(C),$$ which is true....
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### Intuition behind conditioning Y on X in the front-door adjustment formula

I am trying to understand the practicalities of Pearl's front-door criteria to estimate causal effects under an unobserved unconfounder. First, to give context to the question we have a graph that ...
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### What relationships do spatial statistics share with time series analysis?

Spatial statistics is often discussed in tandem with time series. How are the two related? Do they share methodologies? Do overlap in assumptions or conditions of data?