Questions tagged [conditional-independence]

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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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Is my current understanding of the conditional independence assumption in bayesian inference correct?

Standard bayesian inference asssumptions: model $M=\{p(\cdot | \theta) | \theta \in \mathcal{T} \}$ as a prior, choose a distribution $\pi$ on $\mathcal{T}$ then assume observations are generated as ...
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Conditional Independence between two Independent Variables

I have the following question: if $x$ and $y$ are independent random variables/events, $z$ depends on $x$ and $z$ depends on $y$. Are $x$ and $y$ conditional independent on $z$. My first intuition is ...
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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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How to calculate by hand dependent variables sampled from a Frank Copula?

Let's consider this code: ...
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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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1 answer
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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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1 vote
1 answer
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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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1 answer
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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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1 answer
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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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3 answers
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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 Testing using Fisher

I would like to prove the conditional independence of two categorical variables ("a","b") given a third categorical variable "z". I need some help in explaining the ...
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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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Find conditional independence between the attributes of a categorical dataset

I have a high dimensional data set. I used feature selection method to reduce the dimensionality of the dataset. Originally, the dataset has 120 attributes which I minimized to 80 attributes after ...
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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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Degrees of freedom in conditional independence of categorical variables

I'm new to the concept of degrees of freedom and so far I only saw example of degrees of freedom in settings that are different from the one of this question therefore I decided to post my doubts here....
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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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Conditional indepencies in Bayesian network. Redundant edges in structure learning?

I am confused about whether I can have connected 'triangles' in BN assuming that all variables are observed (no missing values). I see that 'bnlearn' software and other softwares too give me a network ...
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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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1 answer
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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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Does conditional independence mandate zero partial correlation?

A and B are conditionally independent random variables given C. Does this necessarily imply partial correlation coefficient of A and B adjusted for C (rho_AB.C) is zero? Can you give an example if its ...
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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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4 votes
1 answer
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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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1 answer
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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 $...
5 votes
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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 ...
1 vote
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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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Show conditional independence of $W_{i}$ and $X_{i}$ given propensity score

In Recent Development in Econometrics while discussing estimation methods based on the Propensity Score, Imbens and Wooldridge state that for any binary variable $W_{i}$, and any random vector ${X_i}$ ...