# 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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### 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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### 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 ...
1answer
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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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### Naive bayes example by hand

Given the following data ...
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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### 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{...