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Questions tagged [conditional-independence]

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Conditional independence problem for poisson random variables

I have this problem: Let $X = V + W$ and $Y = V + Z$ where $V, W, Z$ are independent Pois($\lambda$) random variables. I found that $Cov(X, Y) = Var(V) = \lambda$ It now asks to find whether $X$ ...
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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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1answer
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d-separation in Bayes Network vs separation in undirected graph

I've been teaching myself about d-separation and am trying to answer the following question. Given the graphs below, write down all conditional independence relationships involving the random variable ...
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50 views

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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1answer
52 views

Expectation of potential outcomes formula

In Mostly Harmless Econometrics, the author uses the following identity to derive an estimator for the causal effect: $$E \left[ \frac{Y_i D_i} {p(X_i)} \right] = E \left[Y_{1i} \right]$$ where: $...
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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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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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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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Naive bayes example by hand

Given the following data ...
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1answer
46 views

Conditional independence and joint distributions in graphical models

I'm reading Deep Learning by Ian Goodfellow and Yoshua Bengio and Aaron Courville. In chapter 3 about graphical models, to reduce the model complexity, we assume that certain conditional independence ...
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1answer
57 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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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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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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Do we assume conditional independence to prove it?

Sorry if this is a silly question but it comes from reading a book on directed graphical models. They show algebraically that two variables $x$, $z$ are conditionally independent given $y$. It says: ...
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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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1answer
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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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1answer
55 views

Independence and conditional distribution

In a problem that I'm solving I find that: "Let data (yi,xi) be sampled randomly from a two-dimensional distribution such that y|x is N(ɑ,x^2σ^2)". Are y and x i.i.d? maybe just identically ...
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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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1answer
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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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synthetic datasets without any dependecies between features

I have to create a synthetic data set without any dependencies between features so that this equation should be hold. I thought about to take simply several Gaussians or randomisers, each would be ...
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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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1answer
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Joint densities and conditional independece

Let us assume the joint density $p(x,y,z)$ is factorized as $p(y)p(z|y)p(x|z)$. Hence, $x \perp y|z$. Now, the posterior distribution of z is: $p(z|x,y)=\frac{p(x,y,z)}{p(x,y)}$, where $p(x,y)=\int p(...
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Is the example distribution conditionally independent?

I am studying for my machine learning exam and i have problems with basic conditional probability. I have to solve the following exercise: The formular for conditionally independence is $P(Y|x) = \...
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1answer
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Assuming n variables are conditionally independent given y, how do I compute p(y | x_1,…,x_n)?

Referencing this question, I know that if $x_1$ and $x_2$ are conditionally independent given $y$ (big assumption), then $$P(y | x_1,x_2) = \frac{P(x_1,x_2 | y)P(y)}{P(x_2 | x_1)P(x_1)}$$ $$ = \frac{...
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1answer
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graphical models: integration rules over conditional prob distns

Suppose we have a graphical model over three RVs $a,b,c$  [![enter image description here][1]][1] whose conditional independence structure gives  $$ p(a,b,c) = p(a)\,p(c|a)\,p(b|c), \quad p(a,b)= \...
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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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199 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 ...
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1answer
349 views

Conditional Independence Example

Is there a canonical example of data which are conditionally independent? In other words, $X_1,\ldots,X_p$ are mutually independent given $Y$. This is the foundational assumption of the naive Bayes ...
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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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1answer
458 views

Conditional independence: conditioning on an empty set of random variables

Is $X \perp\!\!\!\perp Y$ a conditional independence, arguing that the independence is conditioned on an empty set of random variables? If so, does that mean that an unconditional independence is ...