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Questions tagged [conditional-independence]

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Testing for conditional independence: What's the correct way?

My goal is to check if two variables $X$ and $Y$ are conditionally independent given $Z$. For simplicity, let's assume the joint distribution is multivariate normal. In this case, we can compute ...
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2answers
54 views

Order of Conditional Independence Tests

I'm studying the PC algorithm for learning the structure of a Bayesian Network. One of the steps refers to performing several rounds of conditional independence tests of increasing order, zero, first,...
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Building independence maps (I-maps) from data

I am just getting into Bayesian networks, and I am having a hard time understanding how this algorithm works: http://pgm.stanford.edu/Algs/page-79.pdf (The algorithm is from Probabilistic Graphical ...
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Improving the Naive Bayes classifier performance through decorrelation?

I was wondering if it is possible to improve the performance of the Naïve Bayes classifier by decorrelating the data. The Naïve Bayes assumes conditional independence of the features given some class $...
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Sampling with fixed probability from two different distributions. How is the sample distributed?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mu$ be a probability measure on $(\mathbb R,\mathcal B(\mathbb R))$ $X$ be real-valued random variable on $(\Omega,\mathcal A,\...
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Creating multivariate regression model out of multiple univariate models

A bunch of ML regression models are defined only for predicting the value of a single variable. Or have standard implementation that are only for the univariate case. For example support vector ...
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1answer
65 views

conditional probability involving mixed variable types

I'm trying to answer the following question A defective coin minting machine produces coins whose probability of heads is a random variable $T$ with PDF $f_{T}(p) = 1+\mathrm{sin}(2\pi p)$ if $p \in ...
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Conditional independence of four variables

I read an argument about variables $A,B,C,D$ that are not mutually independent. It supposed the existence of $P(A,B,C,D)$ where $A \perp B \mid \{C,D\}$ and $C\perp D \mid \{A,B\}$ (with $\perp$ ...
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Compatibility of conditional and marginal independence assumptions

I want to know if two independence assumptions, as illustrated below, would go together or not. Consider I have 4 variables, A,B,C,D. Can the following two independence assumptions co-exist? $A \...
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Conditions Mutual Information and Confounding Effect

Given that conditional mutual information (CMI) I(A,B |C) is the information shared between A, and B given C, does this consider the confounding effect -if any - that C introduces? In other words, ...
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How would I find $P(X \ne Y)$ given independent conditional probability mass functions?

Suppose that $W$ has a discrete uniform distribution on $\{1,\cdots,n\}$. Further, suppose that given $W=w$, the random variables $X$ and $Y$ are independently identically distributed geometric random ...
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correlation between signals

I have some sensor measurements (traffic speed cameras) which are deployed all over a city and totalling about 10000. I have data from them for the last 8 years with a fairly decent temporal frequency ...
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If outputs are independent why can we drop the condition on other outputs' inputs?

In a book the author is trying to explain why we cannot assume independence among outputs but rather conditionally independence. He gives this example: Imagine we had values of all Olympic years ...
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Relationship between strict exogeneity and exogeneity in Time Series ADL models

I am currently learning about ADL models in Time series regression. The textbook notes down two types of exogeneity: Strict exogeneity and exogeneity. Exogeneity is defined as $$E(u_t|X_t,X_{t-1},...)...
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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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Conditional Distribuiton

When trying to solve the following question I have to take into account the dependency of $\alpha_2$ to $b_1$, $b_2$ and $S$? Since, apparently, $\alpha_1$ only depends on $\theta_1$ e $\theta_2$. Any ...
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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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Conditional density under conditional indepencence?

Let $X,Y,Z$ three random variables such that the joint density can be factorized as $$f(x,y,z) = f(x \mid z) f(y\mid z) f(z).$$ This is, I am assuming conditional independence of $X$ and $Y$ given $Z$....
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Derivation of the formula for the probability of a class, given conditionally independent attributes

The following is a formula that finds the posterior probability of a class (i.e. yes or no) given four conditionally independent attributes: $$P(c|X) = P(x_1|c)\cdot P(x_2|c)\cdot P(x_3|c)\cdot P(x_4|...
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Do GGMs model partial correlation or conditional independence or both?

Post that says partial correlation != conditional dependence/independence: https://www.quora.com/Does-a-partial-correlation-of-0-imply-conditional-independence Post above points to following paper: ...
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103 views

Naive bayes example by hand

Given the following data ...
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Why dependencies would cancel while paramterizing CPD?

Consider that in CPTs of a Bayes Net(structure is fixed) one dependence is fix(parameters are set for $P(X|Y)$) and we are parameterizing other CPDs, is it possible that the other parameters would ...
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1answer
25 views

Conditional Independence

I have a joint probability, which factors as follows: $P(A,B,C,D) = P(A,B) \cdot P(C|A) \cdot P(D|B)$ So I know that $C$ and $D$ are independent given $P(A, B)$ right? I want to infer $P(A,B|C,D)$....
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Is joint conditionally independent equivalent to marginally conditionally independent?

Heading ##I am wondering whether these two properties are equivalent: $X$ is conditionally independent of $Y$ given $Z$ $X$ is conditionally independent of $Y$ given $a^T Z$, $\forall a \in R^p$ ...
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209 views

Chi-Squared Statistic to test for conditional independence

How can compute the chi-square statistic between the random variables X and Y given an evidence variable Z? I want to test if X & Y are conditionally independent given Z. Assume all X,Y, Z are ...