Questions tagged [conditional-independence]
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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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Unconfoundedness vs CIA vs selection on observables
I just had a quick question about the CIA, conditional unconfoundedness and secelction on observables only. Do these three terms mean the exact same thing or are there differences between the three?
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Markov blanket - probability derivation
Is this correct reasoning?
Let $x_i$ be a variable in a Bayesian Network and $\text{MB}(x_i)$ denotes its Markov blanket.
Let us note that:
$$
p(x_i \mid \text{MB}(x_i)) \propto p(x_i, \text{MB}(x_i))....
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Why implied Conditional Independencies of mediator and confounder are the same?
I am trying to understand why the impliedConditionalIndependencies function of the rethinking package returns the same value for ...
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Testing implications of Conditional Mean Independence
In some empirical studies as a validation exercise some people regress some variables on the variable of interest controlling for key control variables. The reason for doing this I think comes from ...
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Notational confusion about conditional independence in Pearl 2009
First I read this definition which introduces $X$, $Y$ and $Z$ as sets of random variables.
Definition (Pearl 2009)
Let $V = \{V_1, V_2, \ldots \}$ be a finite set of variables. Let $P(\cdot)$ be a ...
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Meaning of Conditional Independence Between Sets of Variables
I am reading about the so called "global Markov property" of a Markov random field and this is defined as the conditional independence between two sets of random variables given a third set
$...
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Ratio between expectation of maximum of $n$ and $n-1$ IID random variables
Let $X_1, ..., X_n$ be iid random variables. Define $Z_n = \max(X_1, ..., X_n)$. Can we lower bound
$$\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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For Bernoulli schemes and Bernoulli process how to prove that events are independent?
I've understood that for a Bernoulli scheme and for a Bernoulli process (Markov chain with two vertices) the probability of the future doesn't depend on the present (neither of the past, as it was ...
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dependence for variables that are not d-separated
I need to show that for a linear SEM having X->Y<-Z means that X and Z are dependent conditional on Y. For a linear SEM with errors that have finite variances this is doable, but for a model ...
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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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conditional randomization for computing stratum causal effect
This is related to Robin's What If Causal Inference Sec 4.2. Let $Y^i(i=0,1),A,L,V$ be potential outcomes(binary),treatment(binary),covariate(binary) and stratum(binary). Stratify subjects by $V$ and ...
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Gaussian process based likelihood ratio test for conditional independence
I'm exploring alternative approaches to conditional independence tests and wanted to know if it's possible to use create a test statistic from a likelihood ratio test with different data.
Can I create ...
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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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"Predictive dependence" between two variables
Given two random variables $X$ and $Y$, it is natural to use the conditional entropy $H[Y|X]$ to quantify the extent to which knowing $X$ decreases the uncertainty about $Y$. However, consider the ...
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BayesNet Independence
For BayesNet, can anyone explain how we can check the independence between the set of random variables? e.g. $\{B, D\} \perp \{G, I\} | A?$
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Not necessarily conditionally independent = dependent?
After concluding the d-separation procedure (ancestral graph -> moral graph -> removing directed links), I am left with two nodes that are connected and a conclusion that they are "not ...
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What is the most elegant way to express conditional independence on a line graph?
Consider a Markov graph
$$x_1 -x_2-x_3-...-x_t$$
In such a graphical model, we have the conditional independence property $x_{s-1} \perp x_{s+1:t} | x_s \;\forall\; x=2,...,t-1$ and $x_{1:s-1} \perp ...
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How to show mathematically whether the following conditional relationships hold?
In the following Bayesian network, the variables $ x_{i} $ are mutually independent (let's assume that these are the positions of $N$ boats). The variables $ y_{i,j} $ are distance measurements ...
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Proving Independence due to exchangeability?
I have a set of bernoulli random variables $\{x_i\}^{n}_{i=1}$ and $\{x_{ij}\}_{i< j}$. They have a probability distribution with following conditional independence:
$$P(\{x_i\}^{n}_{i=1},\{x_{ij}\}...
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Bootstrap method for chi squared test of independence
I really need some advice about using the chi-squared test of independence.
I want to use the bootstrap-chi-squared method for conditional independence testing. The problem is that the DOF is really ...
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If $p(A,B,C,D) = p(A,B) \cdot p(C,D)$, then is $p(A \mid B,C,D) = p(A \mid B)$?
Given the discrete random variables $A,B,C,$ and $D$, if
$$
p(A = a,B = b,C = c,D = d) = p(A = a,B = b) \cdot p(C = c,D = d) \ \forall a,b,c,d
$$
then is
$$
p(A = a \mid B = b,C = c,D = d) = p(A = a \...
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Tests with null hypothesis of dependence
Let's say I have a set of variables $\mathbf{V}$ and I want to study conditional dependence between two of them $A, B\in \mathbf{V}$ by conditioning on a set $\mathbf{Z}\subseteq\mathbf{V}$. In other ...
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Directed graphical models and independence (exercise)
Context: this is Ex. 1 in these notes http://www.stat.cmu.edu/~larry/=sml/DAGs.pdf .
The exercise asks to prove that, given a directed graphical model associated to a DAG (directed acyclic graph) $G$:
...
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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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