Questions tagged [conditional-independence]

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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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Bayesian error in binary classification when covariates are conditionally iid

In the setting of this problem, $\eta(\vec{x})$ is $P(Y=1|\vec{X}=\vec{x})$, $Y \in {0,1}$, $X \in R^d$. Being the true probability know, the classification rule is simply $\eta(\vec{x})>0.5 \...
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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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62 views

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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1answer
37 views

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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1answer
43 views

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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In which cases GNB is worse than logistic regresion?

I am training and testing two models on the same dataset: a logistic regression and a gaussian naive bayes (sklearn's with the default parameters). The dataset is the ...
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12 views

Where do the “semantics” of a Bayesian network come from?

On Bayesian Networks, Ghahramani (2001) says: A node is independent of its non-descendants given its parents. This point is fundamental enough that Ghahramani calls it the “semantics” of a Bayesian ...
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27 views

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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55 views

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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1answer
88 views

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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1answer
74 views

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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1answer
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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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1answer
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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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1answer
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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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3answers
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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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1answer
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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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Bayesian structure learning: how to identify z as a collider in x-z-y structure?

In BNSL(Bayesian Network Structure Learning) problem, we are asked to learn a DAG(Directed Acyclic Graph) over a randon variable set $U$, given samples of the underlying distribution of $U$. The ...
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Shouldn’t we say independent given the distribution?

In statistics we often deal with iid random variables: independent identically distributed. But if we don’t know the distribution (say we still know the support is {0, 1}), and we get a sample x1, say ...
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1answer
69 views

show that A and B are independent

Given $P(A,B,C,D) = P(A)P(B)P(C|A,B)P(D|C)$ show that $P(A,B) = P(A)P(B)$ First one can write $P(A,B,C,D) = P(A,B)P(C,D|A,B) $ hence $P(A,B)P(C,D|A,B) = P(A)P(B)P(C|A,B)P(D|C)$ $P(A,B) = \frac{P(A)...
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Identifiability of a probability given a set of conditional independence statements and distributions

I am seeking help for finding papers demonstrating the identifiability of a probability given a set of conditional independence statements and a set of probability distributions. More specifically, I ...
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70 views

Improving the Naive Bayes classifier performance through decorrelation?

I was wondering if it is possible to improve the performance of the Naïve Bayes classifier by decorrelating the data. The Naïve Bayes assumes conditional independence of the features given some class $...
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1answer
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How would I find $P(X \ne Y)$ given independent conditional probability mass functions?

Suppose that $W$ has a discrete uniform distribution on $\{1,\cdots,n\}$. Further, suppose that given $W=w$, the random variables $X$ and $Y$ are independently identically distributed geometric random ...
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1answer
120 views

Conditional independence problem for poisson random variables

I have this problem: Let $X = V + W$ and $Y = V + Z$ where $V, W, Z$ are independent Pois($\lambda$) random variables. I found that $Cov(X, Y) = Var(V) = \lambda$ It now asks to find whether $X$ ...
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1answer
121 views

d-separation in Bayes Network vs separation in undirected graph

I've been teaching myself about d-separation and am trying to answer the following question. Given the graphs below, write down all conditional independence relationships involving the random variable ...
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Sampling with fixed probability from two different distributions. How is the sample distributed?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mu$ be a probability measure on $(\mathbb R,\mathcal B(\mathbb R))$ $X$ be real-valued random variable on $(\Omega,\mathcal A,\...
2
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1answer
148 views

Expectation of potential outcomes formula

In Mostly Harmless Econometrics, the author uses the following identity to derive an estimator for the causal effect: $$E \left[ \frac{Y_i D_i} {p(X_i)} \right] = E \left[Y_{1i} \right]$$ where: $...
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Conditional density under conditional indepencence?

Let $X,Y,Z$ three random variables such that the joint density can be factorized as $$f(x,y,z) = f(x \mid z) f(y\mid z) f(z).$$ This is, I am assuming conditional independence of $X$ and $Y$ given $Z$....
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Creating multivariate regression model out of multiple univariate models

A bunch of ML regression models are defined only for predicting the value of a single variable. Or have standard implementation that are only for the univariate case. For example support vector ...
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115 views

Derivation of the formula for the probability of a class, given conditionally independent attributes

The following is a formula that finds the posterior probability of a class (i.e. yes or no) given four conditionally independent attributes: $$P(c|X) = P(x_1|c)\cdot P(x_2|c)\cdot P(x_3|c)\cdot P(x_4|...
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Naive bayes example by hand

Given the following data ...
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1answer
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Conditional independence and joint distributions in graphical models

I'm reading Deep Learning by Ian Goodfellow and Yoshua Bengio and Aaron Courville. In chapter 3 about graphical models, to reduce the model complexity, we assume that certain conditional independence ...
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1answer
544 views

conditional probability involving mixed variable types

I'm trying to answer the following question A defective coin minting machine produces coins whose probability of heads is a random variable $T$ with PDF $f_{T}(p) = 1+\mathrm{sin}(2\pi p)$ if $p \in ...
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1answer
29 views

Conditional Independence

I have a joint probability, which factors as follows: $P(A,B,C,D) = P(A,B) \cdot P(C|A) \cdot P(D|B)$ So I know that $C$ and $D$ are independent given $P(A, B)$ right? I want to infer $P(A,B|C,D)$....
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88 views

Do we assume conditional independence to prove it?

Sorry if this is a silly question but it comes from reading a book on directed graphical models. They show algebraically that two variables $x$, $z$ are conditionally independent given $y$. It says: ...
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Conditional independence of four variables

I read an argument about variables $A,B,C,D$ that are not mutually independent. It supposed the existence of $P(A,B,C,D)$ where $A \perp B \mid \{C,D\}$ and $C\perp D \mid \{A,B\}$ (with $\perp$ ...
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Order of Conditional Independence Tests

I'm studying the PC algorithm for learning the structure of a Bayesian Network. One of the steps refers to performing several rounds of conditional independence tests of increasing order, zero, first,...
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1answer
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Independence and conditional distribution

In a problem that I'm solving I find that: "Let data (yi,xi) be sampled randomly from a two-dimensional distribution such that y|x is N(ɑ,x^2σ^2)". Are y and x i.i.d? maybe just identically ...
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Building independence maps (I-maps) from data

I am just getting into Bayesian networks, and I am having a hard time understanding how this algorithm works: http://pgm.stanford.edu/Algs/page-79.pdf (The algorithm is from Probabilistic Graphical ...
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1answer
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Is joint conditionally independent equivalent to marginally conditionally independent?

Heading ##I am wondering whether these two properties are equivalent: $X$ is conditionally independent of $Y$ given $Z$ $X$ is conditionally independent of $Y$ given $a^T Z$, $\forall a \in R^p$ ...
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Compatibility of conditional and marginal independence assumptions

I want to know if two independence assumptions, as illustrated below, would go together or not. Consider I have 4 variables, A,B,C,D. Can the following two independence assumptions co-exist? $A \...
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1answer
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Joint densities and conditional independece

Let us assume the joint density $p(x,y,z)$ is factorized as $p(y)p(z|y)p(x|z)$. Hence, $x \perp y|z$. Now, the posterior distribution of z is: $p(z|x,y)=\frac{p(x,y,z)}{p(x,y)}$, where $p(x,y)=\int p(...
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1answer
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Is the example distribution conditionally independent?

I am studying for my machine learning exam and i have problems with basic conditional probability. I have to solve the following exercise: The formular for conditionally independence is $P(Y|x) = \...
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Assuming n variables are conditionally independent given y, how do I compute p(y | x_1,…,x_n)?

Referencing this question, I know that if $x_1$ and $x_2$ are conditionally independent given $y$ (big assumption), then $$P(y | x_1,x_2) = \frac{P(x_1,x_2 | y)P(y)}{P(x_2 | x_1)P(x_1)}$$ $$ = \frac{...