All Questions
Tagged with conditional-independence conditional-probability
44
questions
2
votes
1
answer
118
views
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
votes
2
answers
2k
views
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
6
votes
1
answer
608
views
Dropping condition from conditional probability
Consider 3 random variables $X$, $Y$ and $Z$. Under which conditions would we have $P(X\mid Y,Z) = P(X\mid Z)$?
- 305
4
votes
2
answers
114
views
Cumulative distribution of Gaussian conditional independent random variables
Suppose X, Y, Z are three jointly Gaussian random variables and X and Z are independent given Y. For example, take three r.v. from a OU process. Here is some R code:...
- 41
3
votes
1
answer
144
views
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
0
votes
0
answers
41
views
Clarify, with example, completeness conjecture by Pearl and Paz
I was going through Probabilistic Reasoning In Intelligent Systems by Judea Pearl. A completeness conjecture (for which no complete proof is there as yet, but has been found to be true generally, as ...
2
votes
1
answer
113
views
Quick way to determine the different independence assumptions
This question is different than my previous question in that I'm asking sort of a "meta" question.
Here's two graphical models (a Belief Network and a Markov Network):
I would like to ...
- 71
1
vote
1
answer
145
views
Determining unconditional independence in Markov Networks
I would like to know whether $E \perp\kern-5pt\perp A $ in the following Markov Network and would like to know if my reasoning is correct:
So, since this is a Pairwise Markov Network, it factorizes ...
- 71
1
vote
1
answer
290
views
Checking for conditional independence in graphical models
I would like to know whether $B \perp\kern-5pt\perp C | D,A $ and $D \perp\kern-5pt\perp A | B,C $ in the following two graphical models and would like to know if my reasoning is correct:
For the ...
- 71
1
vote
1
answer
50
views
Is $C \perp\kern-5pt\perp D | A $ for the two graphical models? [duplicate]
I would like to know whether $C \perp\kern-5pt\perp D | A $ in the following two graphical models and would like to know if my reasoning is correct:
For the left model (Belief Network), here's my ...
- 71
2
votes
1
answer
66
views
Is $B \perp\kern-5pt\perp C | A $ for the two graphical models?
I would like to know whether $B \perp\kern-5pt\perp C | A $ in the following two graphical models and would like to know if my reasoning is correct:
For the left graphical model, which is a Belief ...
- 71
0
votes
1
answer
83
views
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,
...
- 71
1
vote
3
answers
296
views
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:
$$P[AA] = P[A] \neq ...
1
vote
1
answer
316
views
Condition on two random variables
I'm trying to set up the proper assumptions for a proof I'm working on:
Given that $P(A|e) = P(A)$ and $P(A|c,e) = P(A|e)$, can we prove that $P(A|c)=P(A)$?
I understand that A is independent of e and ...
- 13
1
vote
0
answers
22
views
What is the most elegant way to express conditional independence on a line graph?
Consider a Markov graph
$$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 ...
- 565