Questions tagged [conditional-independence]

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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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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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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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5 votes
2 answers
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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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Markov networks which are disconnected

I'm reading Kollers book on PGMs. Some of her examples to show the breaking down of theorems around independencies for non positive distributions involve either empty Markov networks or very sparse ...
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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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Joint and Conditional Distributions in Bayesian Network

In such a graphical model, how can I express the conditional probabilities $p(x_4|x_1,x_2)$ and $p(x_4,x_5|x_1,x_2)$? My work: $p(x_4|x_1,x_2) = p(x_1,x_2,x_4) / p(x_1,x_2) p(x_1,x_2) = p(x_1)*p(x_2|...
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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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2 votes
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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 ${X_i}$ ...
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2 votes
1 answer
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Computing probability distributions in the two envelope problem

I am trying to understand the resolution to the two envelope problem. While I am still working my way through it and so far the progress has been good I am stuck at a claim that one of the sources ...
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How can a random variable be independent of a member of its minimal Markov blanket?

Consider the following Bayes network of random variables on some probability space: The local Markov property asserts that any variable is independent of its non-descendants given its parents. Here, $...
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Is Pearson's chi-squared test of independence conditional on marginal distributions?

The Wikipedia page on Pearson's chi-squared test states that a difference to Fisher's exact test is that the latter makes the "assumption of fixed marginal distributions". I assume that ...
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Is there any work on, given a set of conditional independences, build the graphical model?

The graphical model Represents probabilistic independence. Given a set of conditional independence assumptions, how to find the probabilistic graphical model that maximizes some metrics (e.g, minimum ...
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How to combine two conditional CDFs

I am trying to reason about the following scenario: Let us have three random variables: $X$, $Y$, $Z$. $Y$ is independent of $Z$. Let us also have the following CDF's: $$F_X, F_{X \mid Y}, F_{X \mid Z}...
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Conditional independence, partial correlation

In my work, I am modelling graphs by measuring the zero- and first-order conditional independence between the variables. That is, if there are three variables, say $A$,$B$ and $C$, an edge between the ...
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2 votes
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How to reason about independence of combinations of events?

Suppose sets $A , B,$ and $D$ are independent. Is it guaranteed that $A \cap B^c \cap D$ is independent from $B^c \cup D^c$? Isn't $B^c$ ($B$ complement, or $B$ not happening) giving me ...
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How does one decides if conditional Poisson response is valid for count data?

Suppose I have count data grouped in equal time intervals as a dependent variable. Often a Poisson regression is a better suited GLM model then, say, conditional Gaussian. Due to my little training ...
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Are the factors in a time series factor model independent over time?

Consider the time series extension to the standard factor model: $$X_t = \Lambda F_t + e_t, \qquad t = 1, 2, \ldots, T$$ where $X_t$ represents \the vector of observations at time $t$, $F_t$ ...
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2 votes
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Causality: Models, Reasoning, and Inference: Notation Question Concerning Graphoids

$\newcommand{\ci}{\!\perp\!\!\!\perp\!}$On page 11 of the book in the title, Pearl introduces the Dawid notation for conditional independence: $(X\ci Y|Z)_P$ if and only if $P(x|y,z)=P(x|z)$ for all ...
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Is this even possible?

How is this possible: If $P (Z|Y, X) = P(Z|Y)$ AND $P(X|Y,Z)= P(X|Y)$ How can these two be equal: $$P (Z|Y,X) = P(X|Y,Z)$$
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Given random variables $X,Y,Z$, under what conditions is $P(Y|X)=P(Y|X,Z)$?

From this link, where the statement is given for events and not random variables, I gather that for random variables $X,Y,Z$, $P(Y|X)=P(Y|X,Z)$ only if $P(Y,Z|X)=P(Y|X)P(Z|X)$? Does this imply that $Y$...
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Are two coin flips conditionally independent if we know that the coin is biased towards heads?

Suppose Alice (A) and Bob (B) each flip the same, potentially-biased coin. Then, P(A=H) < P(A=H | B=H), because Bob's flip increases our suspicion that the coin is biased towards heads. Now ...
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Bishop PRML Question 8.10: d-separation [closed]

I have trouble with solving the second part of question 8.10 from Bishop's PRML (attached as image). I tried several things. Here's my latest attempt: \begin{align} p(a, b, d) &= \int p(a)p(b)p(...
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bayesian network conditional independence test

In the book: Bayesian Networks With Examples in R, the author does this independence test: As I see it, this works both ways, we test if travel is independent of education likewise if education is ...
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How can I calculate the likelihood of my data given that two variables in my data are dependent?

I have a data set D that has 3 variables X, Y, Z, where each variable has 100 samples and have a Normal distribution. I am interested in calculating $p(X \not\!\perp\!\!\!\perp Y | D)$ and $p((X \not\!...
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Proving conditional independence using a bayesian belief network / factorization

I have a bayesian belief network with 4 binary variables $A, B, C, D$. I now need to proof that for joint probability distributions factorized according the Bayesian network given below the ...
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