Questions tagged [conditional-independence]
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Why does dimensionality affect significance and effect size in a Full Conditional Independence (FullCI) test?
The 2018 Runge et al paper titled "Detecting causal associations in large non linear time series datasets" describes the PCMCI method. It compares the new PCMCI method with another method ...
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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}(\...
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Conditional independence in BUGs/JAGs?
I am trying to create a hierarchical model in BUGs. I am actually attempting to implement this is Nimble, but I suspect that a JAGs implementation will be informative.
To attempt to reduce my problem ...
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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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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 ...
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Conditional likelihood, conditional independence and joint independence
Consider a sequence of data samples generated from $n$ independent random vectors $(X_1, Y_1), (X_2,Y_2), (X_3,Y_3) ...$
$$D = (x_1,y_1), (x_2,y_2), (x_3,y_3) ...$$
Where $(X_i, Y_i)$ - is a random ...
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Conditional Independence: Equivalent Conditions
Let $X_1$ and $X_2$ be random variables, and $R(X_1)$ be a function of $X_1$. Here are two statements:
(a) $X_1\perp\!\!\!\!\perp (X_2, Y) \mid R(X_1) $
(b) $X_1\perp\!\!\!\!\perp Y \mid \{R(X_1),X_2\}...
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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 ...
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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 ...
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Join distribution of independent random variables that aren't conditionally independent
I am asked to give an example for a joint distribution of three random variables, $U$, $V$ and $W$, where $U$ and $V$ are (unconditionally) independent but are NOT conditionally independent given $W$.
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Necessary assumptions for a permutation test of conditional independence
Consider a case in which you want to know if two variables (X, Y) are independent conditional given a set (C) of other variables. A recent paper (Shah, R. D., and J. Peters. 2020. The hardness of ...
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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:
$P(A \cap B|C) = P(A|C)P(B|C).$
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ANOVA with variables with known (but arbitrary) conditional dependencies
I have a dataset with the following properties:
k > 2 groups
normally distributed
differing variance and sample size between groups
non-independent samples within each group
continuous variable
...
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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 ...
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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 ...
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