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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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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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 ...
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Discrete Bayes Net learning under parameter constraints

What is some relevant research available on estimating the parameters of a Bayes Net (with known structure) when there are known constraints on conditional and marginal probabilities? For example, ...
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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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277 views

Find Conditional probability distribution given conditions - Bayes Network

I'm taking a course on bayesian statistics and I'm having trouble with one assigment, it goes like this: Construct an example in which two variables have a common effect, and the presence of one of ...
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1 answer
655 views

How to express Bayesian Network or Markov Random Field using deep learning

Bayesian Nework and Makov random field are instances of general probabilistic graphical model. Is it possible to express Bayesian Network or Markov Random Field using deep learning? or in general to ...
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question about notation of Bayes theorem

i was reading Yarin's thesis on Bayesian neural networks and MC-dropout and , at section 2.1 at page 18, I stumbled upon this formula $$ p(\bf{w}|\bf{X},\bf{Y})= \frac{p(\bf{Y}|\bf{X},\bf{w})p(\bf{...
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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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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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1 answer
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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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1 answer
276 views

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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Derive Probabilities from Bayesian Network

I have given the following Bayesian network. I already know the fundamentals, for example, pairwise dependencies and how to calculate the total joint probability: $$P(M,B,S,C,H) = P(M)\cdot P(S|M)\...
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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 net probability question

I've made this Bayes net based on a problem and I'm trying to find the probability of W but I'm stuck. I know I probably have to use Bayes theorem backwards through to find $P(W)$, but I'm not sure ...
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601 views

Generate dataset from Bayesian network

Given a Bayesian network consisting of n random variables. What is the algorithm to generate the dataset from the given structure and associated conditional ...
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1 answer
397 views

How to train a Bayesian network with Bernoulli switch variable?

I model my problem as a simple V-structured Bayesian network. There is an $outcome$ variable, the binary $switch$ variable, and some environment features $X$. All the variables are observed during ...
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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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754 views

In this Bayesian network, where does this posterior probability come from?

I'm reading Building Intelligent Interactive Tutors (Woolf, 2009) on student models for ITSs. On page 261, the author presents an example for a simple Bayesian network ($S \rightarrow E$), where $S$ ...
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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 ...
2 votes
1 answer
23 views

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 ...
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1 answer
157 views

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 ...
1 vote
1 answer
222 views

How can you draw samples from a multidimensional time series?

I'm aware of MCMC methods such as Metropolis Hastings and friends, but these methods assume a stable posterior distribution. Is there a way to draw samples from a multidimensional timeseries? For ...
1 vote
1 answer
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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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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 ...
7 votes
2 answers
384 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 ...
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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 ...
1 vote
1 answer
709 views

Markov blanket conditional distribution derivation

I am trying to derive the formula for the conditional distribution for a variable in a Bayesian network: $$p(x_j|x_{-j})=p(x_j|x_{pa(j)})\prod_{k\in ch(j)}p(x_k|x_{pa(k)})$$ I understand D-separation ...
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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 ...
1 vote
1 answer
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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 ...
1 vote
1 answer
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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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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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1 answer
37 views

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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60 views

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 ...
2 votes
2 answers
432 views

How is the log likelihood calculated for bayesian networks?

In structure learning, there are score-based methods which rely on information criteria such as BIC or AIC. BIC, specifically, is defined as: $$ BIC = k \ln(n) - 2\ln\left(\hat{L}\right) $$ Where $k$ ...
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156 views

Understanding parameter sharing within Bayesian Networks

I am learning about probabilistic graphical models and am a bit confused by the idea of parameter sharing. In the image below, I have been told that the parameters of time slice 0 are copied to time ...
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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 ...
2 votes
1 answer
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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 ...
1 vote
1 answer
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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 ...
2 votes
1 answer
281 views

What does it mean for a variable to block a path between other two variables?

What does it mean for a variable $Z$ to "block" the path between variables $X$ and $Y$ in a causal model? What is the formal definition of a "block", and how can I intuitively understand this concept? ...
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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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