# Questions tagged [bayesian-network]

A Bayesian network is a probabilistic directed acyclic graph. Nodes represent random variables in the Bayesian sense (observable or unobservable); edges represent conditional dependencies between nodes.

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### Propagation of change through a Bayesian Network with continuous variables

I have created a Bayesian network from a number of continuous variables using the bnlearn package in R. One of the things that the package provides is the regression coefficients of each node vs. its ...
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### What is Gaussian approximation for the variance of a function?

In Orre 2000, the author provides an asymptotic approach to computing the variance of information component and conditioned posterior distribution. In part 2.2 weights and information components So ...
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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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### Application of Bayesian Networks to tabular data

I have been going through some tutorials regarding Bayesian Networks, but i have yet to see them applied to tabular data, i.e. a dataset. I have created this dummy example to experiment, and attempt ...
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### Compute joint distribution from fully connected factor graph

I've searched but can't find a meaningful answer (I'm more a software dev than math person, so I'm likely misunderstanding something) Assume I have several variables (A, B, C, D, E, etc) from a ...
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### How is backdoor criterion used in practice?

Is the backdoor criterion applicable only for "learning" in a causal model (i.e. for estimating the causal effects between variables) or must it also be used when running that model, as in ...
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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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### Clarification on Bayesian ensembling

I was reading a paper https://arxiv.org/pdf/2007.06823.pdf and at the end of page 3 the author presents the technique called "ensembling" for the estimation of the expected outputs and the ...
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### Conditional independence proof

I want to prove that $\mathbb{P}(X|U,P) = \mathbb{P}(X|U) \implies \mathbb{P}(X|U,P,T) = \mathbb{P}(X|U,T)$ Where all the letters denote random variables. I'm not sure that this is right, but it seems ...
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### Bayesian Network edges and their meaning

I was wondering why are we not able to conclude dependencies between nodes/variables from edges in a Bayesian Network, while we are able to conclude independence between nodes/variables from the ...
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### What justifies the multiplication step in the proof of the front-door adjustment?

$\newcommand{\doop}{\operatorname{do}}$ The proofs of the front-door adjustment that I've read take three steps: Show $P(M|\doop(X))$ is identifiable Show $P(Y|\doop(M))$ is identifiable Multiply the ...
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### How to compare Bayes nets and Deep learning networks from performance vs explainability tradeoffs?

Machine learning models with higher performance are often based on more complex algorithms and therefore lack interpretability or explainability and vice versa. How to compare the two important ...
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### What are the priors in Bayesian Neural Network?

So I have knowledge of Bayesian statistics/econometrics, but I only recently found out about Bayesian Neural Networks. I was wondering what the priors are in a Bayesian Neural Network? And is it ...
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### BayesNet Independence

For BayesNet, can anyone explain how we can check the independence between the set of random variables? e.g. $\{B, D\} \perp \{G, I\} | A?$
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### where to find the reference Bayesian networks of XMLBIF(.xml) format?

The XMLBIF(.xml) format is XML Belief Network Interchange File Format, which is one of the standard formats for the storage and manipulation of the Bayesian networks. It is widely used by some ...
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### what does the alpha symbol represent in the standard equation used for inference by enumeration

can anyone tell me what the alpha means in the following equation (used for inference by enumeration)?
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### Gaussian Bayesian Networks versus Multiple Linear Regression

I have a set of predictors, $X_1$ to $X_5$, and four response variables, $Y_1$ to $Y_4$. All variables are continuous and distributions are either approximately Gaussian or can be easily transformed ...
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### Hybrid Bayesian Network with Continuous Parents and Ordinal Child Node (Softmax Function)

Upon reading about hybrid BNs with discrete child nodes and multiple continuous parents, I came across the possibility of using the softmax (multinomial logit) function (below) in order to query ...
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### What is the t deterministic transformation in Bayesian Neural Networks?

I am learning Bayesian neural networks and have one question about the t deterministic transformation function, which builds the bridge between the neural weights and the variational parameters. Known ...
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### Finding the maximum likelihood parameter estimates for a given bayesian network

I am studying the book by Adnan Darwiche (Modelling and Reasoning with Bayesian Networks), specifically Chapter 17. I am unsure about how to proceed with this exercise. Since A leads to B and B leads ...
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### How to show mathematically whether the following conditional relationships hold?

In the following Bayesian network, the variables $x_{i}$ are mutually independent (let's assume that these are the positions of $N$ boats). The variables $y_{i,j}$ are distance measurements ...
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### Bayesian Interpretation of Deep Ensembles

I was wondering if training a neural network in the deep ensemble setting can lead to a network with a posterior vs. a point estimate architecture? Recently there have been discussions over the ...
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### How do you compute the messages in a Bayesian network?

I'm just starting to figure out simple Bayesian Networks. This tutorial (https://towardsdatascience.com/belief-propagation-in-bayesian-networks-29f51fdc839c) has been the most accessible so far, but ...
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### How are Judea Pearl's Bayesian Nets different from Google's Causal Impact?

At a high level, how are these two approaches, similar and different? I understand that both use Bayes rule, however, I'm unclear on how they differ. Causal Impact uses structural time series to ...
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### How to calculate the number of parameters in a network analysis and what that means for sample size

I'm hoping to conduct a network analysis (Ising model then later add LASSO regularization) on a biobank sample with a lot of data. Something like 2,000+ variables and 90k+ patients. It'd be nice to do ...
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### Understanding Bayesian Hierarchical Model in Practice

I have a Bayesian hierarchical model with datapoints $y_{ij}$ which are samples from distributions with parameters $\theta_j$. For each distribution parameter $\theta_j$, there are $n_j$ datapoints ...
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### Causal Inference in a Bayesian Network with unobservable backdoor and no frontdoor

Just like the Bayesian network shown above. I want to identify the average treatment effect(ATE) from Smoking to Lung Cancer. If Genetics is observable, I can easily identify the ATE with Pearl's ...
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### Conditional indepencies in Bayesian network. Redundant edges in structure learning?

I am confused about whether I can have connected 'triangles' in BN assuming that all variables are observed (no missing values). I see that 'bnlearn' software and other softwares too give me a network ...
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### Conditional independence tests not respecting d-separation

Wrong example, please refer to the second example I just tried to model a Bayesian network composed of 3 variables as follows $A\sim N(0,1)$ $B\sim A + N(0,1)$ $T\sim A + B + N(0,1)$ In the DAG ...
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### Initializing the nodes of a bayesian network

I am doing some reading about Bayesian networks and how to represent them with a DAG. I have a question about how to initialize the distribution properties of the nodes. Say there is a Bayesian ...
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### I-map Bayesian Network, Practical Explanation

I am having diffculty understanding the concept of an I-map in the context of Bayesian Networks. According to the PGM textbook by Koller & Friedman, an I-map is essentially a set of conditional ...
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### In bayesian approach, what is the difference between full posterior and MAP [duplicate]

Consider a classic machine learning problem, which we want to solve using NN. And suppose that we want to use bayesian learning for that. In the bayesian approach the posterior is described as follows:...
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### Proving Equivalence between Multivariate distributions and Gaussian Bayesian Networks

I am studying Probabilistic Graphical Models by Daphne Koller. In Chap 7, the authors say the following. I can't convince myself of the highlighted part. Induction typically has a statement for n, ...
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### Pearl's Front-door and Back-door

I've encountered lots of causal inference terms and jargons (under the Neyman-Rubin potential outcome framework), and I had a question regarding how Pearl's DAG restrictions relate to ignorability and ...
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