Questions tagged [bayesian-network]

For questions related to Bayesian networks, the generic example of a directed probabilistic graphical model. Includes dynamic Bayesian networks, e.g. Hidden Markov Models (HMMs) and Kalman Filters. For applications of Bayesian networks in any field, e.g. machine learning. NOT for general questions about Bayes' theorem, Bayesian statistics, conditional probabilities, networks, or graph theory.

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Conditional Probability about Bayesian Network

Given the above Bayesian Network ($A,B,C,D$ are events), how can I prove the following equality? $$ \begin{align} P(D|A) &= P(D|B \cap C)P(B|A)P(C|A)+ \\ &\ \ \ \ \ \ P(D|B \cap C^c)P(B|...
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minimal active trail

I'm looking at the question One thing i'm confused on is whether it's necessary that $X_{i-1}$ or $X_{i+1}$ should be given? Define the shorter trail to be the one which replaces $X_{i-1}, X_i, X_{i+...
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Linear Gaussian Networks

I'm reading through a two slides on Linear Gaussian Networks and i'm confused as to how they've calculated variance. In one case they claim (1) $\mathrm{cov}(X_i,X_j) = \sum_{k \in Pa_j} w_{jk} \...
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How has this posterior been calculated via marginalization?

I'm reading a paper, and it has a very simply generative model, which is represented by this . They calculate the posterior P(A|X) in a way I don't understand, though. It doesn't look like a ...
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How to find variance of predictions for Probabilistic Power flow using gpml tool box?

Context: I am trying to model Probabilistic power flow in Gaussian Process framework using gpml toolbox. Question: The problem with the command [ymu1, ys1,f,s] = gp(hyp, @infGaussLik, meanfunc, ...
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Conditional independence in Bayesian networks: A simple example

Consider a probability space $(\Omega,\mathscr{F},\mathbb{P})$. Suppose $A:(\Omega,\mathscr{F})\rightarrow (\mathbb{A},\mathscr{A})$ $B:(\Omega,\mathscr{F})\rightarrow (\mathbb{B},\mathscr{B})$ $G:(\...
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Compute the Posterior from a Hierarchical Bayesian Graph

Suppose that I have the following Bayesian network: I want to calculate $p(\omega | \mathbf{x})$, where $\mathbf{x} = [x_{1}, \ldots, x_{N}]$ is my observed data. My distributions are: $$ \mathbf{x}...
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Estimating the Parameters in a simple Bayesian Graph

Here I have the following Bayesian Graph: In other words, $\alpha$ and $\theta$ are parameters, while $\pi, \mathbf{z}, \mathbf{x}$ are random variables. From the graph, I know that: $f(\mathbf{x}, \...
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Uprooting a node in graph

I was reading this paper https://arxiv.org/pdf/1305.5506.pdf . On page 9, it is written that $$\sum_{x. - a.}P(x. - a.|\hat{a}) = 1$$ Here, $G$ is a graph whose nodes are $\underline{x}$. and $\...
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When does maximum of the Bethe energy function corresponds to true marginals?

I have a question and I would be quite grateful if you could answer. It is known that Belief propagation converges to stationary points of Bethe energy function. Is it known that the maximum of 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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Bayesian Network Enumeration

I want to calculate P(L) for the given Bayesian Network. The solution that I am presented with by the lecturer is 0.170 My calculation path is as following. Since we know that in a Bayesian network ...
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Unintuitive result while working with a temporal Bayesian Network

I am working on a temporal Bayesian Network toy problem using BayesFusion GeNIe Software. I have a node (Case_24 in the figure) that models the state (0 or 1) of a time-dependent variable. At every ...
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Algebraic proof of conditional independence in a Bayesian network

I have a Bayesian network shown as $a \to b \to c \leftarrow d \leftarrow e$. I want to prove $a \perp e$. It's easy to show $b \perp d$ since $$P(b,c,d)=P(c|b,d)P(b)P(d)=P(c|b,d)P(b|d)P(d).$$ So ...
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Understanding normalization in Bayes net

I can't seem to wrap my mind around the concept of normalization. I am hoping these examples will clarify my understanding. If I have a variable A (which has 3 values eg something like A = Sunny, ...
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How to solve Bayes' probabilistic network problem?

Given the following Bayesian Probabilistic Network, let's say I am trying to find the probability of P(!FO|HB). I understand basic Bayes theorem, but not sure how to use it here.
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Graphoid axioms properties doesn't make sense to me.

I am studying some concepts about d-separation through the book [1], however, I am not understanding the intuition behind the axiomatization of graphoids. If anyone can help me with a little ...
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How to calculate the probability of L from this Bayesian Network

This is a follow up to a previous question posted. I am now working on calculating P(~B|~F). The Bayesian Network: So far, I have: ...
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Bayesian network probability truth table

I am working on a bayesian network problem and have been given the following table of data: ...
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Probability and Bayesian Networks: Set of nodes reachable from X via trails active in G, given Z

I am self studying the book Probabilistic Graphical Models. In a question I am asked to prove: The algorithm Reachable(G, X,Z) returns the set of all nodes ...
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Calculating Joint Probability Distribution from Bayesian Network

Apologies in advance if this is considered an easy topic. I am absolutely mired and feel so defeated. I am working with the following Bayesian Network: I am being asked to compute the following: <...
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inference by enumeration on Bayes graph

a graph with known conditional prob between nodes, i.e. P(P2 | P1), P(P2 | ¬P1) are all known: ...
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conditional independence on Bayes graph

I am confusing on conditional independence on Bayes graph. a graph: P6 ↓ P1 → P3 → P4 → P5 ↓ P7 Please kindly let me know if ...
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D-Separation in a Bayesian Network

I am working with the graph below: I know the three cases of d-separation are below (taken from here): I need to find ALL pairs of nodes separated by {A} and <...
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What really the message means in the Belief propagation

I am trying to understand the Belief propagation, and I have question about the message: the message is: $m_{ij}(x_j)$ to reresent the message from node i to node j Does $x_j$ mean the attribute ...
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Got stuck at trying to figure out what the single shot at inference for Variational Autoencoder should be

Let's say you have an already trained Variational Autoencoder where the parameters are $\phi, \theta$ for the recognition and generative models respectively. Let's also assume you have the following ...
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Probabilities in simplified Bayesian (?) network

I have the following simple network (sorry for the link, I cannot insert pictures yet) with nodes and conditional probabilities attached to the edges, and I am looking for an algorithm to efficiently ...
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Computing a conditional probabilty based on a directed graph

I am self-studying graphical models and I have come across the, what seems to be famous, "Student Network". I have included a picture of this graph below. There is a question that asks to compute $$p(\...
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What is a complement of conditional independence?

I newly study Baysian ball algorithm. There is a V-structure X$\rightarrow$ Z $\leftarrow$ Y, where X, Y, Z are random variables, which implies $$ P(X,Y,Z) = P(X)P(Y)P(Z|X,Y). $$ I also found that ...
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How to prove d-separation implies conditional independence?

All of the materials I see online just state it as fact. I don't see it as obvious at all. I use this definition of a Belief network. And this is the definition of d-seperation from the textbook:
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Notation on distribution over finite sets

I'm puzzled about the meaning of $\Delta S$ in the following statement: At each time period, $t \in N$ and conditional on the realization of state $\theta$, agent $i$ observes a private signal $\...
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Conditional Independence Relations for $X_1\leftarrow X\rightarrow X_2$

Let $X$ be a random variable, and let $X_1:=g_1(X)$ and $X_2:=g_2(X)$. Does it hold that $X\perp \!\!\! \perp X_1 | (X_1, X_2)$? (This statement is made in the proof of Proposition 1 in the appendix ...
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conditioning on the source or target variables in d-separation?

In Pearl's Causality - Models, Reasoning and Inference (2009), he defines d-separation as follows: Let $X\perp\!\!\!\perp Y |Z$ mean "$Z$ d-separates $X$ from $Y$". But there seems to be a weird ...
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Gibbs Sampling for the “Clutter Problem”

i am trying to use gibbs sampling to to estimate a Posterior of the clutter problem but I can't get my head around how to do it. The problem is states as follow: We sample a value µ from a N(0,a), ...
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Is this a structural causal model with structural minimality, but not causal minimality?

Background I am solving problems from "Elements of causal inference" by Peters, Janzing, Schölkopf (2017, MIT Press). The book is available in open access (link). I'm stumped by problem 6.57b. Since ...
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conditional independencies regarding Markov condition

Maybe someone can help me out a bit. I am learning for an exam and got some additional exercises. Sadly, without solution, so I can not check my solutions. Sadly I can not ask someone else as well, ...
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1answer
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find d-seperation in DAG

i got a question regarding DAGs and d-seperation within the theme Bayesian Networks. In Task d) i am supposed to check for the following d-seperations. But i am struggeling. My other excercises ...
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Nonparametric (at leasat partial) Bayesian programming?

https://en.wikipedia.org/wiki/Bayesian_programming expects assumptions about the form of the probability distributions and hences - those distributions are parametric. But is there non-parametric ...
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Conditional Independence of Non-Descendants in Bayesian Network

Suppose we have the following order of random variables: $X_1, X_2, \ldots, X_n$, from which a Bayesian Network is constructed. Let $p(X)$ denote the parents of random variable $X$. Consider an ...
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1answer
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Trying to Understand Iterative Conditional Modes (ICM)

I am looking at pseudo code for ICM What is the point of the outerloop at line 3? It doesn't seem like it's being used at all?
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1answer
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Bayesian Network and conditional probability - concrete example

bayesian network picture Abbreviations: A ... Visited Asia S ... Smoker T ... Has tuberculosis L ... Has lung cancer B ... Has bronchititis E ... Has either tuberculosis or lung cancer X ... ...
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Variable Elimination where order is Alarm and then earthquake

Suppose I have the following: And I want to calculate the following: $P(B=true|J=true,M=true),A-E$ I have found online this link with examples about Variable Elimination. where here in my case ...
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Showing that $E,B$ are independent

Suppose we have the following scenario: And I want to tell if $B$ and $E$ are independent. It looks like there are independent because it seems that they don't have a common parent, but I need ...
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1answer
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Simple question: Posterior for one coin flip

Just wondering if I assume a uniform prior:$𝑝/(1−𝑝) $, and I have a unfair coin with head probability = $p$. If I flip 3 times and observed 2 heads, how doe I compute the posterior? The answer is ...
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bayesian network conditional distribution encoding

I want to know if I am barking up the wrong tree with my bayesian network model. I have a data set, and a DAG that looks like this: (a->b),(b-c),(c->d),(c->e). For variables d & e, I want to ...
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Decision Network Expected Utility

Suppose I am given the following decision network, with $t\in\{T_n,T_t,T_s\}$ being the decision: $f\in\{F,\bar{F}\}$, $h\in\{H,\bar{H}\}$, $s\in\{S,\bar{S}\}$. Suppose I am given $P(f)$, $P(s|f)$, $...
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In probability, what “prior knowledge” means??

So i have this equation, that is a Bayesian Score Metric: In this context, what "prior knowledge" means?
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Help with Cooper and Herskovits (1992) Bayesian Score Metric

Im studying the Chickering (1996) "Learning Bayesian Networks is NP-Complete", available in this link: http://maxchickering.com/publications/lns96.pdf And i have some questions about the following ...
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Learning Bayesian Networks is NP-Complete (Chickering Proof)

I'm studying the NP-Complete of Learning Bayesian Networks by the paper of David Maxwell Chickering, available in this link: http://people.cs.pitt.edu/~milos/courses/cs3710/readings/...
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How to find causal direction in a maximally weighted spanning tree

I have a dataset from which I have constructed a network based on the order of dependency - KL Divergence aka Mutual Information. I would like to find the causal direction i.e., which node causes ...

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