Questions tagged [conditional-independence]
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Distribution of $\max_i \bar{X}-X_i$
Let $X_1, \ldots, X_n$ be i.i.d. random variables from the standard normal distribution and let $\bar{X} = \frac{1}{n}\sum_{i=1}^n X_i$ be their sample mean.
I'm interested in the distribution of the $...
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Conditional expectation function and causal inference
!For the question itself skip to the last paragraph!
It is my understanding that iff we have a model of the form $$Y = m(X) + e$$ and $E[e|X] = 0$ we know that $m(X)$ is the conditional expectation ...
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A question of "elementary imsets" in an ADMG
In [The m-connecting imset and factorization for ADMG models] (https://doi.org/10.48550/arXiv.2207.08963), it was mentioned the notation of an "elementary imset". The definition of ...
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How do I control for pscore?
I see that people usually implement pscore matching or control for pscore fixed effects. Why do I need to do pscore matching? Why can't I just include pscore as a continuous variable in my regression ...
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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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Bayes Rule with conditional independence of two tests based on a common blood sample
I have the following scenario of Bayes updating with which I struggle quite a bit.
Imagine we are interested in the probability that a given person has a disease $D$. We perform two different tests $...
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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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Testing for conditional independence with nonlinear relationships
I am reading about the IC and IC* (Inductive Causation) algorithms for discovering DAGs from observations. The first step of the algorithm is for each pair of variables a and b, search for a set of ...
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Non-parametric tests to compare conditionally independent groups
I want to compare two groups using the Mann Whitney U test (also known as the Wilcoxon rank sum test) per this description: https://sphweb.bumc.bu.edu/otlt/mph-modules/bs/bs704_nonparametric/...
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Conditional independence statements for probabilistic motivation for linear regression
So the motivation for using the squared loss in linear regression can be written as the following (I think):
Assume $\{(\mathbf{x}_i, y_i) \mid i = 1, \dots n\}$ are repeated independent samples from ...
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Why implied Conditional Independencies of mediator and confounder are the same?
I am trying to understand why the impliedConditionalIndependencies function of the rethinking package returns the same value for ...
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Notational confusion about conditional independence in Pearl 2009
First I read this definition which introduces $X$, $Y$ and $Z$ as sets of random variables.
Definition (Pearl 2009)
Let $V = \{V_1, V_2, \ldots \}$ be a finite set of variables. Let $P(\cdot)$ be a ...
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Ratio between expectation of maximum of $n$ and $n-1$ IID random variables
Let $X_1, ..., X_n$ be iid random variables. Define $Z_n = \max(X_1, ..., X_n)$. Can we lower bound
$$\mathbb{E}[Z_{n-1}] \geq (1-f(n))\mathbb{E}[Z_n]$$
Using some $f(n)$. I am mainly interested in ...
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Does this independence property hold?
Let $x \sim N(\mu_x,\Sigma_x)$ and $v \sim N(0,\Sigma_v)$ be independent multivariate Gaussian random vectors, and let $$y = Ax + v$$ for some square matrix $A$ such that $y \sim N(A\mu_x, A\Sigma_xA^...
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Is treatment conditionally independent from outcome in Single Experiment Design?
I'm reading this slides.
At slide 10 there is written that in "Single Experiment Design" we assume "Randomization of treatment", that is:
$ \{ Y_i(t,m),M_i(t') \} \perp T_i \lvert ...
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