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Questions tagged [marginal]

The marginal distribution refers to the probability distribution of a subset of variables contained in a joint distribution.

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Name for special marginal distribution

Given $P(X_1, \dots, X_N)$ is there a name for the following two marginal distributions? The marginal, $P(X_n)$, including only the $n$-th variable The marginal, $P(X_1, \dots, X_{n-1}, X_{n+1}, \...
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Generating marginal posteriors from MCMC output in two-factor model

Quick summary: if I have a MCMC sample of the posterior distribution of two factors and their interactions, can I marginalize out one factor simply by concatenating the posterior samples from each ...
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80 views

A Bernoulli mixture model with a Dirichlet prior on the parameters

Let's $x_1,...,x_N$ be a set of observation coming from the following generative process: $$ \boldsymbol{\theta} \sim \text{Dirichlet}(\boldsymbol{\alpha})\qquad\boldsymbol{\theta},\boldsymbol\alpha\...
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sampling from joint distribution to recover marginal distribution

I'm going through Bayesian Core and have gotten stuck at this remark on page 233: " A first remark that motivates the use of the Gibbs sampler is that, within structures such as $$ \pi(x_1) = \int \...
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Integrating a variable out of a distribution - The Graph Elimination Algorithm

Consider the graphical model below for 6 binary variables. It defines a joint distribution $\begin{align} p(x_1,x_2,x_3,x_4,x_5,x_6) &= p(x_1) p(x_2|x_1)p(x_3|x_1)p(x_4|x_2) p(x_5|x_3)p(x_6|x_2,...
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variables that are normaly distributed but their joint distribution is not multivariate normal with ρ = 0.5 [duplicate]

can you give me an example or explain me how to find one. It can be with copulas
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8 views

Marginalisation when conditioning on the Fourier transform of a random variable

I am looking to sample from a distribution $p(y)$ defined by the following expectation: $$p(y) = \mathbb{E}_{p(u)} \left[ p(y|u) \right]$$ Both $p(u)$ and $p(y|u)$ are multivariate Gaussians: $$p(u)...
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AIC / BIC for Model Seleciton in Copula Model

I'm trying to select the distributional model of 30 marginals (which are restricted to have the same distributional family) in a copula model. However, I therefore get 30 Likelihood/AIC/BIC values for ...
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Estimation of joint errors

Suppose we have multiple normal random variables, each which is a normal distribution with it's own mean but joint variance: $$X_i \sim N(\mu_i, \sigma^2)$$ Now suppose we collect data from these ...
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Derivation for mixed distribution, Poisson-Lindley

I want to derive the Poisson Lindley Distribution. $$ f_x(x|\lambda) = \frac{\lambda^{x-1}}{(x-1)!}e^{-\lambda}$$ $$f_x(x|p) = \frac{p^2}{(p+1)}(\lambda+1)e^{-\lambda p}$$ The Distribution of x, $...
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Formula to Calculate Confidence Interval of Marginal Means

What is the formula for the marginal means confidence interval in a regression analysis? Let us have the following regression $Y = b_0 + b_1X + b_2Z + b_3 XZ + b_4 Cov_1 + b_5 Cov_2$ If I get it ...
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Expectation of exponential family distributions

Is there a closed form of the following marginal (one dimensional data) $\pi(\theta|y) = \mathbb{E}_{x \sim \pi_R(x|y)} \pi(\theta|x)$, where both $\pi, \pi_R$ are exponential family distributions?
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Using the pseudo marginal approach for estimating unknown number of Markov chains in this model

I have $K>0$ iid unknown Markov chains $\{X_n^k : n \in \mathbb{N}\}, k=1, \dots, K$ on a discrete state space $S_X = \{1,2,3\}$, each chain runs and gives rise to observations of the form $\{Y_n^k ...
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MAP versus Component-Wise Maximum Marginal

Suppose I have the joint distribution: \begin{align} p(\mathbf{x}) = p(x_1, ... , x_n) \end{align} The maximum a posteriori (MAP) solution is given by: \begin{align} \mathbf{x}_{MAP} = \arg \max p(\...
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Conditional probabiliy and marginalization

These are some simple questions that I typically find unexplained or I guess assumed to be obvious. Say we have two random variables $X$, $Y$ and we write $P(X|Y)$, and then wish to sum over possible $...
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Rewrite joint probability as product of marginals when all the probabilities are $1$ or $0$

I have a question about the possibility of rewriting a joint probability as the product of the marginals when all the probabilities can only take value $1$ or $0$. I start with introducing some ...
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What does this assumption mean regarding equal marginal densities?

Suppose that we have a random variable $\epsilon$ with density $q(\epsilon)$ and $w = t(\theta, \epsilon)$, where $t$ is a deterministic function of a constant $\theta$ and random variable $\epsilon$. ...
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Why $E$ and $F$ are conditionally independent given $C$ and $D$?

Below is a Directed Acyclic Graph (Fig.a). From this figure, it is said that: $E$ and $F$ are conditionally independent given $C$ and $D$. I am confused about it. Let's assume the causal ...
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48 views

Computing a marginal distribution of a joint involving a delta function

Suppose that we have four continuous random variables $x,y,z,$ and $v$ and we want to compute the following integral: $$\int f(x\mid y)f(z\mid x,y)f(v\mid z,x,y)\,dx$$ There are a few conditions: $...
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Why can “one row of this element” in the MCMC output table represent marginal posterior distribution? [duplicate]

Take the table below as an example, which is in Box 8.1 of this book. This table illustrates the converged MCMC output including the first and last five samples for parameters and derived quantities. ...
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From beta distribution to Dirichlet: Estimation of the concentrantion parameters

Searching at least 3 hours about the connection between beta distribution and dirichlet. My problem is: I have a collection of random variables $X_i \sim Beta(a_i, b_i)$. The parameters $a_i$ and $...
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76 views

Inverse distribution of gaussian mixture

In one of the papers I've encountered, the authors propose a copula function $$ c(u_1, \ldots, u_d; \Theta) = \frac{\psi(y_1, \ldots, y_d; \Theta)}{\prod_{j=1}^{d}\psi_j (y_j)}$$ where $\psi(y_1, \...
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37 views

Find marginal distribution

The random vector $(X,Y)$ is uniformly distributed over $$D=\{(x,y): 0 \leq x \leq 2 , 0 \leq y \leq 2-x\}.$$ Find the marginal distribution of the random variables $X$ and $Y$. For the radom vector $...
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Marginalising over Dependent Random Variables

Suppose I have two RVs, $A$, and $B$. Every place I have looked thus far suggests the following for marginalisation, which for me is fine: $f_A(a) = \int_{-\infty}^{\infty} f_{A,B}(a,b)db $. ...
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80 views

Getting marginal means for ANCOVA in R (effect vs emmeans)

I'm a bit new to running GLM models in R, so forgive me is this is a silly question. Context: I'm running an ANCOVA with the goal to control for multiple covariates while understanding the ...
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Logit model: Interpretation of interaction effect with only dummies

I have a large binary logit model (cross-section) with many variables, mostly dummies. From the information on age, I have created 6 dummies: 17-24, 25-34, 35-44, 45-54 (reference group), 55-64 and 65+...
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Constraints on choice of marginal distribution and likelihood

For some time I have been reading into Bishop's Pattern Recognition and Machine Learning. Coming back to some earlier chapters the following got me confused and I am interested where, formally I go ...
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Computing conditional probabilities on multivariate data from covariances

I am struggling to implement some Bayesian algorithm which I hope you may help me with. I am required to compute all probabilities of the form: $$P(Z_i\le z_i\;|\;Z_1=z_1, \dots, Z_{i-1}=z_{i-1}) \;\...
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Phase marginal for a multivariate complex Gaussian density

Suppose $z$ is a random variable taking values in $\mathbb{C}^n$ and admitting the complex Gaussian density $p(z;W) \propto \exp{(-\frac{1}{2}z^*Wz)}$, where $W$ is Hermitian. Let $r$ be the vector of ...
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Distribution of random variable with multinomial sampling distribution and parameters $(n,p)$, where $n\sim$ Poisson with truncation

Suppose you have: $$X\mid N\sim\text{MN}(N,p_1,p_2,\ldots,p_{J})$$ $$N\sim \text{Poisson}(\lambda)$$ What is the marginal distribution of $X$? In this case, the answer is simply this. But... ...
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141 views

Marginalizing a high-dimensional multivariate Gaussian distribution

I have an 11-dimensional multivariate Gaussian, with a covariance matrix with non-zero values in every element. My goal is to marginalize this down to 4 dimensions, but I'm having some computational ...
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svm's margin equation derive question

I hava question about the margin equation $$\frac{a}{||w||}$$ where this equation coming from? I think it substract the $w^{T} +b -a - w^{T}x +b$ but not sure how margin equation derived
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If $\frac{X_a}{X_a+Y_a}$ and $\frac{X_b}{X_b+Y_b}$ are correlated, what about $X_a+Y_a$ and $X_b+Y_b$

Suppose I have four normal random variables:$X_a$,$Y_a$,$X_b$,$Y_b$. $X_a$ and $Y_a$ follow bivariate normal, $X_b$ and $Y_b$ also follow bivariate normal. let $Z_a=\frac{X_a}{X_a+Y_a}$ , $Z_b=\...
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Setting boundaries for calculating $P(Y/X>2)$ choosing $dx/dy$ order [duplicate]

Given two independent variables $X$ and $Y$, with marginal pdfs $f_X(x)=2x, 0 \le x \le 1$ and $f_Y(y)=1, 0 \le y \le 1$, calculate $P(\frac{Y}{X} > 2)$. So this can be written as $P(Y>2X)$, ...
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128 views

Finding the joint CDF using the joint PDF; why can't I do this?

Find the joint CDF of the independent random variables $X$ and $Y$, where $f_X(x)=x/2, 0\le x \le 2, $ and $f_Y(y)=2y, 0 \le y \le 1$. To do this, we can find the CDF separately for each of the ...
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144 views

Computation and interpretation of marginal effects in a GLMM

I am currently working on a GLMM model which uses a Poisson distribution and need to compute and interpret marginal effects from this model. The model outcome consists of a count (COUNT) collected ...
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Numerically/approximately integrating over independent gamma variables

Problem Statement For a problem in biology, I am testing out a joint distribution of the form: $$ X \sim Multinomial(\frac{\theta_1}{\sum \theta_i}, ...,\frac{\theta_n}{\sum{\theta_i}}) \\ \theta_i \...
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1answer
56 views

Is the marginal distribution of a maximum entropy distribution also a maximum entropy distribution?

I am considering under what circumstances maximum entropy distributions are closed under marginalization. The main case I am interested is the following setting: a finite set of discrete random ...
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33 views

Marginal derivation from joint pdf

I have a uniform prior f(Θ) ~ U(4,10) and a uniform 'observation' model f(X|Θ) ~ U(θ-1, θ+1). Their joint pdf is f(X,Θ)=1/12 for 4 < θ < 10 and (θ-1)< x <(θ+1)  and 0 otherwise. If I ...
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Finding the marginal distribution

If $p(y) =N(0,\Gamma) $ and if $p(x|y) =N(\Lambda y, \psi I) $where $I$ denotes the identity matrix. Also, we have the conditions: $\Gamma_{ij} =0$ for $i \neq j$ (diagonal) $\Lambda \Lambda^T = I$ ...
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Help with PCA Question

The conventional model for probabilistic principal component analysis has a standard normal latent $\vec{y}$ and a loading matrix $\Lambda$: $P(\vec{y}) \sim N(\vec{0}, I)$, $P(\vec{x}|\vec{y}) \sim ...
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Marginal Effect

If anyone could help me with this question, I would appreciate. in this model salary = B0 + B 1LSAT + B 2LSAT2 + B3log(libvol) + B4rank + u where LSAT is the average standardized test score for ...
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PCA Marginal Distribution

In PCA, if I have a latent $\vec{y}$ with loading matrix $\Lambda$, then the PCA models using: (1) $P(\vec{y}) \sim N(\vec{0}, I)$, $P(\vec{x}|\vec{y}) \sim N(\Lambda \vec{y}, \psi I)$ (2) $P(\vec{y}...
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1answer
39 views

Sufficient conditions for equal marginal medians

Let $X, Y$ be dependent random variables taking values on the same set (either a finite set, an interval or the real line). I'd like to know if there's any condition on $P(Y|X)$ which ensures that $$\...
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40 views

Sum of conditional probabilities = Marginal probability? [closed]

Is the following true?: $P(X=j) = \sum_i P(X=j\mid Y=i)$ Thanks!
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77 views

Marginal Distribution from Bivariate Distribution Matrix

I am doing some practice problems to prepare for my statistics exam, and I just want to know if my reasoning is correct on one problem, and if not, I want to know how I should reason through this. The ...
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2answers
113 views

marginal distribution of normal $\mu$ with inverse gamma prior on $\sigma^2$

(From Bayesian Essentials with R by Marin & Robert) Given that $$\mu | \sigma^2 \sim N(\epsilon, \sigma^2 / \lambda_\mu)\,,$$ and $$\sigma^2 \sim IG(\lambda_\sigma /2, \alpha /2)\,,$$ we want ...
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91 views

Longitudinal Mediation with Inverse-Probabilty Weighting / Marginal Structural Model

Validater, I am currently trying to investigate how mediation analysis can be applied with longitudinal data. I already considered some Structural Equation Models (SEM), e.g. the 3-Wave example of ...
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65 views

Bayesian analysis: comparison of marginal probability distributions

Is it valid to compare mariginal probability distributions from separate Bayesian analyses to infer which scenario is most likely? Specifically, in phylogenetic (evolutionary) analysis, if I ...
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58 views

Having difficulty deciding limits of integration for a joint to marginal pdf

A joint pdf, $f_{X,Y}(x,y)=5$, is given with the following intervals: $-1<x<1$ $x^2<y<x^2+{1\over{10}}$ I am trying to find marginal pdf of $f_Y(y)$ but I am stuck. Trying for hours....