Questions tagged [conditional-independence]
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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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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$$
if $r_{i}=0$, then $Y$ is ...
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Conditional independence proof
I want to prove that
$\mathbb{P}(X|U,P) = \mathbb{P}(X|U) \implies \mathbb{P}(X|U,P,T) = \mathbb{P}(X|U,T)$
Where all the letters denote random variables. I'm not sure that this is right, but it seems ...
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"Predictive dependence" between two variables
Given two random variables $X$ and $Y$, it is natural to use the conditional entropy $H[Y|X]$ to quantify the extent to which knowing $X$ decreases the uncertainty about $Y$. However, consider the ...
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BayesNet Independence
For BayesNet, can anyone explain how we can check the independence between the set of random variables? e.g. $\{B, D\} \perp \{G, I\} | A?$
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Not necessarily conditionally independent = dependent?
After concluding the d-separation procedure (ancestral graph -> moral graph -> removing directed links), I am left with two nodes that are connected and a conclusion that they are "not ...
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What is the most elegant way to express conditional independence on a line graph?
Consider a Markov graph
$$x_1 -x_2-x_3-...-x_t$$
In such a graphical model, we have the conditional independence property $x_{s-1} \perp x_{s+1:t} | x_s \;\forall\; x=2,...,t-1$ and $x_{1:s-1} \perp ...
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How to show mathematically whether the following conditional relationships hold?
In the following Bayesian network, the variables $ x_{i} $ are mutually independent (let's assume that these are the positions of $N$ boats). The variables $ y_{i,j} $ are distance measurements ...
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Proving Independence due to exchangeability?
I have a set of bernoulli random variables $\{x_i\}^{n}_{i=1}$ and $\{x_{ij}\}_{i< j}$. They have a probability distribution with following conditional independence:
$$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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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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