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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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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33 views

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?
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Derivation of the Objective Function for Expectation Propagation

I was reading Expectation Propagation As A Way Of Life and the original paper by Minka Expectation Propagation for Approximate Bayesian Inference and they both say that a fixed point of the EP ...
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19 views

Plotting Vector Embeddings

I am currently working on a project that requires some form of data on the paper. Even though I was able to get some coding done and communicate results, I need some graphs. What I want to do is ...
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Bayesian structure learning: how to identify z as a collider in x-z-y structure?

In BNSL(Bayesian Network Structure Learning) problem, we are asked to learn a DAG(Directed Acyclic Graph) over a randon variable set $U$, given samples of the underlying distribution of $U$. The ...
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Mathematical details in the definition of a Structural Causal Model

Pearl defines (see Causality, Judea Pearl, 2nd ed., Definition 7.1.1) a Structural Causal Model (SCM) as a triple $(\mathscr U, \mathscr V, F)$ where $\mathscr U$ is a set of "exogenous variables," $\...
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Density estimation: the significance of smaller number of samples?

I'm reading Probabilistic Graphical Model: Principle and Techniques by Koller and Friedman. In section 18.5 Bayesian Model Averaging(p825), the author said If we are interested only in density ...
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Computing Gradients for a [-1, 1]-valued RBM

The gradient derivation for a binary-valued RBM with values $\in\{0,1\}$ is well-documented, for example in Goodfellow, et al and here on Cross Validated. However, in some works (e.g., associative ...
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Graph representation of relation between DV and IV

If a condition states that $f(I|Y,Z) = f(I|Z)$ How do I represents this relationship using nodes and edges. $I,Y, Z$ are nodes not sure if the following representation is accurate ? Any suggestions ...
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Why is it difficult to sample from Energy Based Models?

I am trying to understand the following claim which is made in the Deep learning book by Goodfellow et. al about a toy energy-based model (with the apparent motivation of introducing Markov Chain ...
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Goodness of fit for Copula

I was wondering if the graphical methods can be used instead of formal tests such as Cramer-Von Mises test as the GOF for copulas? The scatter plot of pseudo-observations is as follows: The Q-Q ...
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Graph and statistics multiple regression show different things

I’ve run a multiple regression to find out if visual memory declines differenty for the autism group than for the controlgroup. I did find a significant main effect of age, showing that with ...
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Basic doubt on generative models

I am new to statistics and while reading Bishop's book, in the 'Generative models' part 8.1.2. When explaining ancestral sample, he says: To do this, we start with the lowest-numbered node and ...
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Seeing a tree graphical model as a Markov model

I have been doing an exercise task and I encountered an issue. Let's imagine that we have a graphical model(binary tree) as in the image below. To every vertex a rv $X_v$ is assigned which obtains ...
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Marginalization not understood

In the book "Pattern recognition and machine learning" by Christopher M. Bishop, at page 374 The joint distribution corresponding to this graph is again obtained from our general formula (8.5) to ...
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Why do we have to convert Bayes' net to MRF before applying Belief propagation?

is that even correct in the first place? if yes, then why? I've seen articles talking about inference in Bayes' nets, and I've seen others talking about conversion. I don't have the full picture.
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Is A ⊥ B | C where one path active but another inactive?

I'm trying to determine if A ⊥ B | C? I see two paths flowing through elements of C: (1) B <- C - > A (all variables unobserved, active triple; independence cannot be guaranteed for this path) (...
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Parsing and understanding plate notation for topic modeling example?

I'm trying to understand the following plate notation which is used a lot as an example of topic model to introduce variational methods, etc. I wanted to ask if my understanding is correctly depicted ...
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What's the effect of observed variables on PGMs?

For a class we're going over the basics of PGMs. The below example was illustrated to show how (conditional) independence is somehow related to what variables are observed. This concept is perplexing ...
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Why Boltzmann machine is represented as a fully connected graphical model?

The joint factorizes into unaries and pair-wise potentials. If that is the case, then why do we represent it as a fully connected graph? It is misleading and gives the impression that the joint cannot ...
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How to quantify a parent's influence of a node in a Bayesian Network?

Consider a toy bayesian network that models purchase of items at a store. The nodes include: {Brand, Price, Purchased}. It is possible that when you marginalize over price, P(purchase|brand) may ...
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Array size as a random variable in graphical models

Assume that I want to model a mixture of sentences. There are two different sources generating sentences with specific sentence length and word distribution. I had came up with the following graphical ...
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Derivation of Conditional Causal Probabilities

In Causal Inference in Statistics: an Overview, Pearl presents an equation describing distribution from a graphical model presented in figure 3: The author arrives at equality (20) - see image above. ...
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Number of Causal Assumptions in an Overview by Pearl

In the paper Causal Inference in Statistics: an Overview by Pearl, in page 11 (106 if you go by the Journal's indexing), a graphical model is presented in figure 2(a). The text reads (picture below): ...
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Do Conditional Independence Statements in Probabilistic Graphical Models Include Independence Statements?

Been reading Murphy and other books and trying to understand conditional independent statements of a graph: $I(G)$, or $CI(G)$ depending on the reference. They mostly define a conditional ...
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102 views

Optimal graphical model selection for large multivariate categorical datasets

My dataset consists of ~100 columns (i.e. variables) of discrete categorical data. Each variable has between 2 and 50 categories. Ideally, I would use a log-linear model for detection of dependence ...
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What is the benefit of latent variables?

I have a model $p(x)$. How can adding latent variables $z$ help me? What are the main benefits of modelling $p(x, z)=p(x|z) p(z) $ instead of $p(x) $ alone? What would be some examples where modelling ...
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Time series - probabilistic graph model

I am very new to Graph models, this is my first attempt. I have good knowledge of time series analysis. I am looking to build a graph model for a set of time series data - daily stock prices for say ...
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How to check if conditional independence holds from CPDs

These CPDs are presented in the Probabilistic Graphical Models course on Coursera as examples of conditional independence and conditional dependence, respectively. I have a vague idea of why the first ...
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Integrating a variable out of a distribution - The Graph Elimination Algorithm

Consider the graphical model below for 6 binary variables. It defines a joint distribution $\begin{align} p(x_1,x_2,x_3,x_4,x_5,x_6) &= p(x_1) p(x_2|x_1)p(x_3|x_1)p(x_4|x_2) p(x_5|x_3)p(x_6|x_2,...
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How does graphical model of a GP look like?

I'm trying to understand the difference between GP and Markov process. I couldn't find answers on the internet. I figured that graphical models can tell the difference, hence my question.
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Factorization of a completely connected undirected graph with pairwise compatibility functions

Given a completely connected undirected graph (V,E) such that $V=(x_1,\dots,x_5)$ and $E = ((x_i,x_j)_{i<j})$ for $i,j =1,\dots,5$, it is known that there exists a factorization $$ P(x_1,\dots,x_5) ...
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106 views

Energy function of Restricted Boltzmann Machine (RBM)

The energy function for RBM (Restricted Boltzmann Machine) is defined as $$ E(v,h) = -\sum_{i,j} w_{ij} \, v_i \, h_j -\sum_i a_i \, v_i - \sum_i b_i \, h_i $$ with the joint distribution $$ \tag{1} p(...
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Graphical Models Showing Independence Relations

I am attempting an old assignment question on graphical models. I am given a paragraph of information and asked to draw a directed graphical model showing the relationships between the variables and ...
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What does the graph for a convolutional layer look like?

I like to draw out computational graphs when I work on small assignments for computing gradients etc. and was wondering what the graph for a convolutional layer would look like? I can't seem to find ...
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Complete graphs have different v-structures?

From the book "Probabilistic Graphical Models", here says two complete graphs have different v-structures. As I understand, v-structure is like "X->Z<-Y" without edge between X and Y. If so, ...
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d-separation in Bayes Network vs separation in undirected graph

I've been teaching myself about d-separation and am trying to answer the following question. Given the graphs below, write down all conditional independence relationships involving the random variable ...
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1answer
28 views

The implementation of variable-to-factor and factor-to-variable messages?

I read this tutorial on the implementation of CRF and got to know that the normalization is the sum-product message passing. And I also know that there are two types of messages on factor graph: ...
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Why are exogenous variables not used in inference/recognition networks?

I have been working lately a lot on amortized variational inference. That is, doing variational inference using neural networks to approximate a variational distribution (such as in Kingma and Welling ...
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Learning Causal Graph from data

I am quite new to the theory of causal graphs, but from what I understand they are DAG, like Bayesian Networks. Since we have structure learning methods for Bayesian Networks like score based ...
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56 views

The difference between forward algorithm used in CRF and the variable elimination?

I found that in the forward algorithm used in the CRF(and perhaps also in the HMM) the mechanism applied is almost the same as that in the variable elimination(VE) except that the emission ...
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29 views

Computing Local Evidence for Bayesian Networks

I am reading through Kevin Murphy's "Machine Learning: A Probabilistic Perspective" book. I'm interested in understanding how to do exact bayesian inference over a tree structure, as discussed in ...
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137 views

How can I understand the complex regression models?

I can understand how it works when there are two variables in the linear regression model(the shaded circles represent the observed variables, and the white ones the latent variables): We can draw ...
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Action of Fisher information in “Latent Variable Model Selection…”

I'm having trouble understanding the role of the Fisher information matrix in the assumptions of Chandrasekaran et al. 2012. In the paper, the authors define the Fisher information matrix (i.e., ...
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applied papers on probabilistic generative models and inference engines

I am looking for applications papers where people choose some task on which they will do Bayesian inferencing and graphical modeling, and then build an inference engine to infer latent parameters. And ...
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Markov Blanket of two nodes?

I'm trying to solve a question and it has asked for (I feel like I'm confused by everything, would apprecaite some help, thanks), the Markov blanket of {c,d}. From what I've read so far, Markov ...

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