Questions tagged [conditional-independence]
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A and B are independent. Does P(A ∩ B|C) = P(A|C) · P(B|C) hold?
Let $C$, $B$, and $A$ be events in the same probability space, such that $A$ and $B$ are independent and
$P(A \cap C) > 0$, $P(B \cap C) > 0$.
Prove or disprove:
$P(A \cap B|C) = P(A|C)P(B|C).$
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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 ...
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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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Dropping condition from conditional probability
Consider 3 random variables $X$, $Y$ and $Z$. Under which conditions would we have $P(X\mid Y,Z) = P(X\mid Z)$?
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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 ...
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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 ...
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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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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 ...
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Informative Censoring vs. Random Censoring vs. Conditionally Independent Censoring
Let us consider the case of survival analysis with one event. Let $X$ represent a set of covariates about each unit. Let $T_E$ be the (latent) event time of the unit, let $T_C$ be the (latent) ...
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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 ...
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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 $Y$...
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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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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 ...
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Cumulative distribution of Gaussian conditional independent random variables
Suppose X, Y, Z are three jointly Gaussian random variables and X and Z are independent given Y. For example, take three r.v. from a OU process. Here is some R code:...
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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 ...
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