# 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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### 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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### In what ways do conjugate priors compose?

A lot of conjugate priors are known for a lot of likelihood distributions (mostly the exponential family). But most Bayesian models in practice don't just consist of one distribution. Usually, you ...
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### Does the classic burglar,earthquake conditional independence example only apply depending on the state of the known parent?

If I recall, this example forms a graph with (earthquake, burglary) being the two parents of a child (alarm), and the statement is that the two parents are conditionally independent only when child is ...
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### Understanding rare definition of the likelihood function and corresponding posterior from research paper

Reading the paper https://storage.googleapis.com/pub-tools-public-publication-data/pdf/b20467a5c27b86c08cceed56fc72ceadb875184a.pdf i came across a rare definition of the likelihood function that in ...
1 vote
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### BNLearn Bayesian Networks - how is the structure decided?

I'm currently playing around with BNLearn on Python, and creating BN's from Pandas datasets. I'm not really sure why/how the package decides on the structure of the BNs it creates; see for example ...
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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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### Derive and validate a probability using a causal diagram

Given a causal diagram and the conditional probabilities for every adjacent node, I want to calculate a specific probability in two ways - let's call them the "simple" and "alternate&...
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### Impact of sample size when using exponential random graph modeling

I am working on revising a manuscript centered on identifying the drivers behind certain types of student interaction. One critique that I'm having trouble addressing is that they are worried about my ...
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### Calculating Probability in Bayesian Networks

I'm trying to work out how to calculate the outcome probability $\Pr(Y)$ for the following BN: I know the joint probability $\Pr(a,b,c,Y) = \Pr(a) \Pr(b \mid a) \Pr(c) \Pr(Y \mid b,c)$ (which I'll ...
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### Bayes Nets - Understanding Inference

I was wondering if anyone could explain the following to me (from https://lips.cs.princeton.edu/complexity-of-inference-in-bayes-nets/): "Briefly, recall that a Bayesian network consists of a ...
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### How to evaluate validity about Causal Discovery?

When I was trying to do structural learning for causal inference, I perplex for too many causal discovery algorithms(PC, FCI, GES, NOTEARS and more...) There are many structural learning algorithm for ...
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### How to apply the the conditional probability and chain rule formulas in a multivariable case?

I'm learning about Bayesian Fusion and have a question regarding an expression that I wasn't able to prove. It poses the following statement: Assume $y_1, y_2, y_3$ are three observations which are ...
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### What is 'factored models'?

What is 'factored models'? I've seen the term 'factored models' in paper 'The Bayesian Structural EM Algorithm'. But I see it for the first time and it is hard to understand... I tried searching it ...
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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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### How to convert a Gamma GLM to a Pearl's Structural Causal Model (SCM) with exogenous noise terms?

Assume both X and Y are univariate for this. So in Judea Pearl's SCM, you have endogenous variables and exogenous noise. A regular causal Bayesian Network like X->Y where X and Y|X is normal can be ...
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### What is the required sample size for Bayesian Network?

I have used Krejcie & Morgan's table for my survey questionnaire to identify the sample size of my study, and it turns out 384 samples. However, specifically, how many sample size (in minimum) is ...
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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 ...
1 vote
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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 ...
1 vote
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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. $\{B, D\} \perp \{G, I\} | A?$
1 vote
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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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### Bipartite Bayesian Network?

I have a set of $n+\binom{n}{2}$ random variables $\{x_i\}^{i=n}_{i=1}\cup \{x_{ij}\}_{i\leq j}$ and their probability distribution is given as: P(x_1,...,x_n,x_{12},...,x_{ij},...x_{(n-1)n}) = \...
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