Questions tagged [conditional-independence]
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Why does dimensionality affect significance and effect size in a Full Conditional Independence (FullCI) test?
The 2018 Runge et al paper titled "Detecting causal associations in large non linear time series datasets" describes the PCMCI method. It compares the new PCMCI method with another method ...
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Hierarchical models and conditional independence
Suppose that we have a hierarchical model given by (this is Example 4.4.5 of Berger and Casella(2002))
\begin{align*}
X\mid Y&\sim\text{binomial}(Y,p),\\
Y\mid\Lambda&\sim\text{Poisson}(\...
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Conditional independence in BUGs/JAGs?
I am trying to create a hierarchical model in BUGs. I am actually attempting to implement this is Nimble, but I suspect that a JAGs implementation will be informative.
To attempt to reduce my problem ...
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Dependence through an unknown parameter?
Consider an urn from which we sample with replacement. Let $\pi$ represent the proportion of the urn's balls that are black, with the remainder being white.
From a frequentist perspective, each ...
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Bayesian network extracting further conditional independence statements then just from d-separation theorem
Given a Bayesian network $(p,\mathcal{G})$, where $p$ is our joint distribution, and $\mathcal{G}$ is a DAG.
Then by the d-separation theorem we can deduce conditional independence statements, in ...
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Conditional likelihood, conditional independence and joint independence
Consider a sequence of data samples generated from $n$ independent random vectors $(X_1, Y_1), (X_2,Y_2), (X_3,Y_3) ...$
$$D = (x_1,y_1), (x_2,y_2), (x_3,y_3) ...$$
Where $(X_i, Y_i)$ - is a random ...
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Conditional Independence: Equivalent Conditions
Let $X_1$ and $X_2$ be random variables, and $R(X_1)$ be a function of $X_1$. Here are two statements:
(a) $X_1\perp\!\!\!\!\perp (X_2, Y) \mid R(X_1) $
(b) $X_1\perp\!\!\!\!\perp Y \mid \{R(X_1),X_2\}...
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Borel-Cantelli lemma on conditional probabilities
In a probability space $\big( \Omega, \mathcal{F}, P \big)$, suppose $\{E_n\}_{n\in \mathbb{N}} \subseteq \mathcal{F}$ is a sequence of mutually independent events. By Borel-Cantelli Lemma, the ...
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Undirected graphs and implications of independence (Wasserman chapter 18)
In Wasserman's All of Statistics chapter 18, he defines the following undirected graph:
Let $V$ be a set of random variables with distribution $\mathbb{P}$. Construct a graph with one vertex for each ...
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Join distribution of independent random variables that aren't conditionally independent
I am asked to give an example for a joint distribution of three random variables, $U$, $V$ and $W$, where $U$ and $V$ are (unconditionally) independent but are NOT conditionally independent given $W$.
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Necessary assumptions for a permutation test of conditional independence
Consider a case in which you want to know if two variables (X, Y) are independent conditional given a set (C) of other variables. A recent paper (Shah, R. D., and J. Peters. 2020. The hardness of ...
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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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ANOVA with variables with known (but arbitrary) conditional dependencies
I have a dataset with the following properties:
k > 2 groups
normally distributed
differing variance and sample size between groups
non-independent samples within each group
continuous variable
...
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Two random variables X1 and X2 may be partially dependent i.e. X1 is independent of X2 but X2 is dependent on X1?
$X(t)$ is a stochastic process defined on the time interval $(0,T)$. Discretizing the time interval one can specify a random variable $X(t_i)$ as:
$$t_0= 0 < t_1,t_2,...,t_{n−1},t_n=T$$
And may be ...
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Is the following conditional density function equivalent to its unconditional counterpart? [duplicate]
Suppose we have a stochastic series $\{X_t\in\mathbb{R}, t=1,\cdots, T\}$. Further suppose that $G(X_t)=\mathbf{1}_{X_t\geq 0}$ where $\mathbf{1}$ is an indicator function. Can it be concluded that ...
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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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What does it mean for tests to be independent?
When reading about cumulation if type-1 Error, the sentence "for independent statistical tests" occures alot, now I was wondering what this is actually means.
Since tests are also random ...
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Intuition of conditional independence in DAGs
In the DAG above, we have $A$ conditionally independent of $E$ when $C$ and $B$ are observed (that is $A\perp E|B,C$), but not when only $C$ is observed (that is $A\not\perp E|C$). I don't have a ...
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Is A independent of B conditioned on B?
Does $A \perp\!\!\!\!\!\!\perp B | B$ always hold?
Part of me is like yes: if we know the value of $B$, then more information about $B$ can't tell us anything about $A$, and vice versa.
Consider this ...
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What's the relationship between statement "Z causes both X and Y" and "X and Y are independent given Z"?
Suppose I have two statements:
Statement 1: Random variable Z is the common cause for random variable X and Y (Z causes both X and Y)
Statement 2: Random variable X and Y are (conditionally) ...
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What does conditional independence mean semantically?
I've just spent the last 3 hours reading every post, question, Medium article, and textbook entry on conditional independence, and I still don't really understand it. Can somebody explain what it ...
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Implications of violating Bayesian network independence assumptions during inference
Consider the example Bayesian network below where $X \perp \!\!\! \perp Y $ (X is independent of Y).
Assuming that this is the true independence structure of the process that is generating the data, ...
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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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Does conditional independence imply the following identities?
I was reading this paper https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.143.8127&rep=rep1&type=pdf , and it heavily uses conditional independencies for deriving various identities, ...
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Clarify, with example, completeness conjecture by Pearl and Paz
I was going through Probabilistic Reasoning In Intelligent Systems by Judea Pearl. A completeness conjecture (for which no complete proof is there as yet, but has been found to be true generally, as ...
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If I engineer a new feature such that feature C = feature A/feature B, must I drop features A and B from a Gaussian Naive Bayes model?
As the question asks, is it bad data science not to drop the dividend and divisor features when creating a new feature that is their quotient when working with a Naive Bayes model? My understanding of ...
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Quick way to determine the different independence assumptions
This question is different than my previous question in that I'm asking sort of a "meta" question.
Here's two graphical models (a Belief Network and a Markov Network):
I would like to ...
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Determining unconditional independence in Markov Networks
I would like to know whether $E \perp\kern-5pt\perp A $ in the following Markov Network and would like to know if my reasoning is correct:
So, since this is a Pairwise Markov Network, it factorizes ...
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Checking for conditional independence in graphical models
I would like to know whether $B \perp\kern-5pt\perp C | D,A $ and $D \perp\kern-5pt\perp A | B,C $ in the following two graphical models and would like to know if my reasoning is correct:
For the ...
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Is $C \perp\kern-5pt\perp D | A $ for the two graphical models? [duplicate]
I would like to know whether $C \perp\kern-5pt\perp D | A $ in the following two graphical models and would like to know if my reasoning is correct:
For the left model (Belief Network), here's my ...
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Is $B \perp\kern-5pt\perp C | A $ for the two graphical models?
I would like to know whether $B \perp\kern-5pt\perp C | A $ in the following two graphical models and would like to know if my reasoning is correct:
For the left graphical model, which is a Belief ...
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Is it always possible to find a joint distribution $p(x_1,x_2,x_3,x_4)$ consistent with these local conditional distributions?
I am currently studying Bayesian Reasoning and Machine Learning by David Barber, the 4th chapter exercise 4.1 (p 79). The exercise is the following:
Exercise 4.1
Consider the pairwise Markov network,
...
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Prove or disprove : $P[A|B] = P[B]$, the A and B are independent? Is this right?
SOrry if this is extremely easy.
I did the following but I'm a little bit unsure about it:
Let $A=B$, and $P[A]>0$.
Then $$P[A|A] = P[A]$$
But A is not independent with itself:
$$P[AA] = P[A] \neq ...
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Condition on two random variables
I'm trying to set up the proper assumptions for a proof I'm working on:
Given that $P(A|e) = P(A)$ and $P(A|c,e) = P(A|e)$, can we prove that $P(A|c)=P(A)$?
I understand that A is independent of e and ...
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If partial regression coefficient is zero, then $Y$ is independent of $X_i$ conditional on all other regression variables
In a textbook Causal Inference in Statistics - A Primer (p. 81), it says
Given the regression equation $$y=r_{0}+r_{1} x_{1}+r_{2}x_{2}+\cdots+r_{n} x_{n}+\epsilon$$
if $r_{i}=0$, then $Y$ is ...
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Conditional independence proof
I want to prove that
$\mathbb{P}(X|U,P) = \mathbb{P}(X|U) \implies \mathbb{P}(X|U,P,T) = \mathbb{P}(X|U,T)$
Where all the letters denote random variables. I'm not sure that this is right, but it seems ...
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