Questions tagged [conditional-independence]
The conditional-independence tag has no usage guidance.
51
questions with no upvoted or accepted answers
7
votes
0
answers
4k
views
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 ...
2
votes
0
answers
81
views
Ratio between expectation of maximum of $n$ and $n-1$ IID random variables
Let $X_1, ..., X_n$ be iid random variables. Define $Z_n = \max(X_1, ..., X_n)$. Can we lower bound
$$\mathbb{E}[Z_{n-1}] \geq (1-f(n))\mathbb{E}[Z_n]$$
Using some $f(n)$. I am mainly interested in ...
2
votes
0
answers
188
views
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 ...
2
votes
0
answers
74
views
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 ...
2
votes
0
answers
38
views
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 $\perp$ ...
2
votes
0
answers
145
views
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 ...
1
vote
1
answer
35
views
Hierarchical models and conditional independence
Suppose that we have a hierarchical model given by (this is Example 4.4.5 of Berger and Casella(2002))
\begin{align*}
X\mid Y&\sim\text{binomial}(Y,p),\\
Y\mid\Lambda&\sim\text{Poisson}(\...
1
vote
0
answers
8
views
Undirected graphs and implications of independence (Wasserman chapter 18)
In Wasserman's All of Statistics chapter 18, he defines the following undirected graph:
Let $V$ be a set of random variables with distribution $\mathbb{P}$. Construct a graph with one vertex for each ...
1
vote
1
answer
66
views
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 ...
1
vote
1
answer
106
views
Conditional expectation function and causal inference
!For the question itself skip to the last paragraph!
It is my understanding that iff we have a model of the form $$Y = m(X) + e$$ and $E[e|X] = 0$ we know that $m(X)$ is the conditional expectation ...
1
vote
0
answers
66
views
Conditional independence statements for probabilistic motivation for linear regression
So the motivation for using the squared loss in linear regression can be written as the following (I think):
Assume $\{(\mathbf{x}_i, y_i) \mid i = 1, \dots n\}$ are repeated independent samples from ...
1
vote
0
answers
24
views
Implications of violating Bayesian network independence assumptions during inference
Consider the example Bayesian network below where $X \perp \!\!\! \perp Y $ (X is independent of Y).
Assuming that this is the true independence structure of the process that is generating the data, ...
1
vote
0
answers
88
views
Does conditional independence imply the following identities?
I was reading this paper https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.143.8127&rep=rep1&type=pdf , and it heavily uses conditional independencies for deriving various identities, ...
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 ...
1
vote
0
answers
1k
views
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 ...