Questions tagged [conditional-independence]

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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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43 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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60 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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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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38 views

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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32 views

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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44 views

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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25 views

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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112 views

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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Conditional Independence of Observed Counting Process From Conditional Independence of Survival Times

Let $T^*_1, \dots, T^*_n$ be survival times with hazard functions $\alpha_i(t)=f(X_i(t))$ and construct the counting processes $N^*_i(t)=I(t\geq T^*_i)$. Let $(C_1(t),\dots, C_n(t))_{t\in \mathbb{R}_+}...
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Conditional independence of attributes in NB algorithm and independence of levels in Target Encoding

This is not an actual question but I really need what you are thinking about it. I have an advisor, not pretty much knowledgeable about Machine Learning/Deep Learning and Statistics. While we were ...
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129 views

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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30 views

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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24 views

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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d-seperation belief networks

Consider the belief network The question now is to list all nodes which are d-seperated from E given I, G. The formal definition is as follows: any node in $X$ is d-seperated from any node in $Y$ ...
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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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Bayes Factor for each entry in a contingency table?

I have an $m x m$ contingency table. Here is a link to an example contingency table Entry $x_{ij}$ counts the number of times terms $i$ and $j$ have been mentioned in a corpus of documents. I am ...
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68 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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40 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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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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correlation between signals

I have some sensor measurements (traffic speed cameras) which are deployed all over a city and totalling about 10000. I have data from them for the last 8 years with a fairly decent temporal frequency ...
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1answer
77 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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78 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,\...
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105 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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84 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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388 views

Naive bayes example by hand

Given the following data ...
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87 views

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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146 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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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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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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64 views

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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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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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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31 views

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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100 views

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{...
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graphical models: integration rules over conditional prob distns

Suppose we have a graphical model over three RVs $a,b,c$  whose conditional independence structure gives  $$ p(a,b,c) = p(a)\,p(c|a)\,p(b|c), \quad p(a,b)= \sum_c p(a,b,c) = p(a) \sum_c p(c|a)\, p(b|...
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Conditions Mutual Information and Confounding Effect

Given that conditional mutual information (CMI) I(A,B |C) is the information shared between A, and B given C, does this consider the confounding effect -if any - that C introduces? In other words, ...
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265 views

Chi-Squared Statistic to test for conditional independence

How can compute the chi-square statistic between the random variables X and Y given an evidence variable Z? I want to test if X & Y are conditionally independent given Z. Assume all X,Y, Z are ...
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468 views

Conditional Independence Example

Is there a canonical example of data which are conditionally independent? In other words, $X_1,\ldots,X_p$ are mutually independent given $Y$. This is the foundational assumption of the naive Bayes ...
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Testing for conditional independence: What's the correct way?

My goal is to check if two variables $X$ and $Y$ are conditionally independent given $Z$. For simplicity, let's assume the joint distribution is multivariate normal. In this case, we can compute ...