Questions tagged [conditional-independence]
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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 $\max(Y,Z)$?
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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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Does independence implies independence conditionally on max of the data?
Let be $X_1, ..., X_n$ I.I.D. numerical random variables with contiunous density $f$.
Note $M(X) = \max(X_1, ..., X_n)$ their maximum.
Are $X_1, ..., X_n$ independent conditionally on $M(X) = x$ for ...
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AIPW and Cross-fitting (Stanford stat361)
I am reading lecture note (Stanford stat361: https://web.stanford.edu/~swager/stats361.pdf) written by Stefan Wager. At page 23-24 the author states dependent summands become independent after ...
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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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R package to solve Gaussian MLE under conditional independence constraints
Is there any R package or function to solve Gaussian MLE under conditional independence constraints?
Suppose we have $y_i\overset{i.i.d}{\sim}\mathcal{N}(0,\Sigma_{p\times p})$, $i = 1,2,\ldots,n$. We ...
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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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Conditional PDF of conditionally independent variables
Consider three continuous random variables $X$, $Y$, and $Z$. $X$ and $Y$ are conditionally independent given $Z$. What's wrong with the following derivation?
$$
f(x|y) = \int f(x|y,z)f(z) dz = \int f(...
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If $X \in \{1, 2, 3\}$, $Y, Z \in \mathbb{R}$ are random variables, what is meant by $X \not\!\perp\!\!\!\perp Y|Z$?
Let $X \in \{1, 2, 3\}$, and $Y \in \mathbb{R}$ and $Z \in \mathbb{R}$ denote random variables. Suppose that:
$$X \not\!\perp\!\!\!\perp Y|Z.$$
In words, $X$ is not conditionally independent of $Y$ ...
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is it possible that $X_{j}$ and $X_{k}$ are independent of each other conditioning on $Z = f(X_1,\cdots, X_N)$?
Suppose I have $N$ random variables $\{X_j\}_{j=1}^N$ and they are mutually independent. Also, I define $Z = f(X_1,\cdots,X_N)$ for some function $f()$. And I want to know that if it is possible that $...
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Can $X_1$ and $X_2$ be independent conditioning on $X_1+X_2$?
Suppose that $X_1$ and $X_2$ are independent. I wonder if $X_1$ and $X_2$ conditioning on $X_1+X_2$ can be independent as well.
If $X_1$ and $X_2$ are normally distributed, then the above statement is ...
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Conditional independence situation with three variables
Say we have three random variables, $X, Y$ and $Z$, where $X$ is independent of $Z$ (but not $Y$).
Does $E\bigg[ \dfrac{X}{f(Y,Z)} \bigg| Y \bigg] = E[X|Y] * E\bigg[ \dfrac{1}{f(Y,Z)} \bigg|Y \bigg]$?
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Is $X$ and $g(f(X))$ conditionally independent on $f(X)$?
Let $f,g$ be measurable functions and $X$ be a random variable.
Then, is $X$ and $g(f(X))$ conditionally independent on $f(X)$?
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Does mutual independence of X, Y, Z implies conditional independence of X and Y, given Z
Given mutual independence of 3 r.v.s X, Y, Z, can we conclude that X and Y are independent, given Z?
Note that I am interested in case when all 3 r.v.s are mutually independent, not only pair X, Y.
In ...
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Why would we require $p_1 = p_2$ in order for $A_1$ and $A_2$ to be independent? Furthermore, how does $B$ change anything?
I have the following example:
There are two coins, labeled 1 and 2, either or both of which are possibly biased. The probability of a head is
$$P(H \mid \text{coin} \ i) = p_i, \ \ \ \ (i = 1, 2).$$
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