All Questions
Tagged with conditional-independence probability
30
questions
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153
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Dependence through an unknown parameter?
Consider an urn from which we sample with replacement. Let $\pi$ represent the proportion of the urn's balls that are black, with the remainder being white.
From a frequentist perspective, each ...
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9
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Bayesian network extracting further conditional independence statements then just from d-separation theorem
Given a Bayesian network $(p,\mathcal{G})$, where $p$ is our joint distribution, and $\mathcal{G}$ is a DAG.
Then by the d-separation theorem we can deduce conditional independence statements, in ...
- 359
2
votes
1
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118
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Borel-Cantelli lemma on conditional probabilities
In a probability space $\big( \Omega, \mathcal{F}, P \big)$, suppose $\{E_n\}_{n\in \mathbb{N}} \subseteq \mathcal{F}$ is a sequence of mutually independent events. By Borel-Cantelli Lemma, the ...
11
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A and B are independent. Does P(A ∩ B|C) = P(A|C) · P(B|C) hold?
Let $C$, $B$, and $A$ be events in the same probability space, such that $A$ and $B$ are independent and
$P(A \cap C) > 0$, $P(B \cap C) > 0$.
Prove or disprove:
$P(A \cap B|C) = P(A|C)P(B|C).$
- 111
1
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Two random variables X1 and X2 may be partially dependent i.e. X1 is independent of X2 but X2 is dependent on X1?
$X(t)$ is a stochastic process defined on the time interval $(0,T)$. Discretizing the time interval one can specify a random variable $X(t_i)$ as:
$$t_0= 0 < t_1,t_2,...,t_{n−1},t_n=T$$
And may be ...
0
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Is the following conditional density function equivalent to its unconditional counterpart? [duplicate]
Suppose we have a stochastic series $\{X_t\in\mathbb{R}, t=1,\cdots, T\}$. Further suppose that $G(X_t)=\mathbf{1}_{X_t\geq 0}$ where $\mathbf{1}$ is an indicator function. Can it be concluded that ...
- 1,226
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A question of "elementary imsets" in an ADMG
In [The m-connecting imset and factorization for ADMG models] (https://doi.org/10.48550/arXiv.2207.08963), it was mentioned the notation of an "elementary imset". The definition of ...
1
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1
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101
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Does this independence property hold?
Let $x \sim N(\mu_x,\Sigma_x)$ and $v \sim N(0,\Sigma_v)$ be independent multivariate Gaussian random vectors, and let $$y = Ax + v$$ for some square matrix $A$ such that $y \sim N(A\mu_x, A\Sigma_xA^...
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144
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What does conditional independence mean semantically?
I've just spent the last 3 hours reading every post, question, Medium article, and textbook entry on conditional independence, and I still don't really understand it. Can somebody explain what it ...
- 257
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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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1
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3
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296
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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 ...
1
vote
1
answer
88
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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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70
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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 ...
1
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1
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86
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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 ...
- 204
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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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