Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

Questions tagged [conditional-independence]

80 questions
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If $X$ and $Y$ are uncorrelated random variables, then under what condition is $E[X \mid Y] \approx E[X]?$

Suppose $X$ and $Y$ are real random variables that are uncorrelated. Now, uncorrelated does not imply independence, so $E[X \mid Y] \ne E[X]$. However, can they be said to be approximately equal? If so, under what conditions does that approximation…
Bridgeburners
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Can $X_1$ and $X_2$ be independent conditioning on $X_1+X_2$?

Suppose that $X_1$ and $X_2$ are independent. I wonder if $X_1$ and $X_2$ conditioning on $X_1+X_2$ can be independent as well. If $X_1$ and $X_2$ are normally distributed, then the above statement is wrong. I wonder if the statement can be true for…
user1292919
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Are two coin flips conditionally independent if we know that the coin is biased towards heads?

Suppose Alice (A) and Bob (B) each flip the same, potentially-biased coin. Then, P(A=H) < P(A=H | B=H), because Bob's flip increases our suspicion that the coin is biased towards heads. Now instead suppose that we already know that the coin is…
monk
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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 partial correlation of X and Y given Z is by…
Vimal
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Variance of the product of two conditional independent variables

Now I know that the variance of the product of two independent variables $Y$ and $Z$ is:$\DeclareMathOperator{\Var}{Var}$ $\Var(YZ) = \Var(Y)\Var(Z) + \Var(Y)E(Z)^2+\Var(Z)E(Y)^2$ However I would like to know what the variance of the product of…
arezaie
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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, second... What does the order refer to?
Jeremy Voisey
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For normally distributed random variables, if X is independent of Y and X is independent of Z, is X independent of max(Y,Z)?

Suppose $X,Y,Z\sim N(0,\sigma^2)$. $X$ is independent of $Y,$ $X$ is independent of $Z$ (but $Y$ and $Z$ are not independent), is $X$ independent of $\max(Y,Z)$?
Ruth
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Joint distribution where random variables always exist in the same orthant

I am sampling two vectors $x$ and $y$ ($\in \mathbb{R}^n$). First, I sample $x$ from an isotropic Gaussian distribution. Then I want to sample $y$ from the same distribution, but only in the orthant where $x$ exists. For example, if the sampled $x$…
CWC
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Given random variables $X,Y,Z$, under what conditions is $P(Y|X)=P(Y|X,Z)$?

From this link, where the statement is given for events and not random variables, I gather that for random variables $X,Y,Z$, $P(Y|X)=P(Y|X,Z)$ only if $P(Y,Z|X)=P(Y|X)P(Z|X)$? Does this imply that $Y$ and $Z$ being conditionally independent…
Yandle
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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 variables with…
Jen Snow
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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 classifier but it's not clear to me which data…
jjet
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is it possible that $X_{j}$ and $X_{k}$ are independent of each other conditioning on $Z = f(X_1,\cdots, X_N)$?

Suppose I have $N$ random variables $\{X_j\}_{j=1}^N$ and they are mutually independent. Also, I define $Z = f(X_1,\cdots,X_N)$ for some function $f()$. And I want to know that if it is possible that $X_j$ is independent of $X_k$ conditioning on $Z$…
user1292919
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Does mutual independence of X, Y, Z implies conditional independence of X and Y, given Z

Given mutual independence of 3 r.v.s X, Y, Z, can we conclude that X and Y are independent, given Z? Note that I am interested in case when all 3 r.v.s are mutually independent, not only pair X, Y. In other words, is it true that: $p_{X,Y,Z}(x,y,z)…
Eugene Loy
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Computing probability distributions in the two envelope problem

I am trying to understand the resolution to the two envelope problem. While I am still working my way through it and so far the progress has been good I am stuck at a claim that one of the sources makes. The source is this Problem: A wealthy…
MiloMinderbinder
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Is Pearson's chi-squared test of independence conditional on marginal distributions?

The Wikipedia page on Pearson's chi-squared test states that a difference to Fisher's exact test is that the latter makes the "assumption of fixed marginal distributions". I assume that applies to the use of chi-squared as a test of independence,…
A. Donda
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