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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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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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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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“marginal standardisation” for comparing predicted probabilities between two groups? [closed]

I have a doubt regarding to the R package "margins". I'm estimating a logistic model: ...
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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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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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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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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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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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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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Continuous Marginals

What are the conditions under which a continuous multivariate probability distribution in $\mathbb{R}^N$ has continous marginals in all dimensions?
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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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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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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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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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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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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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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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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....
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Calculating odds ratio in Multiple select choice question analysis

I have a multiple response categorical variable (MRCV) and a single response categorical variable (SRCV). Respondents in my survey were presented with 6 choices (6 barriers to access an application) ...
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Resample from data with constraints to the marginal distribution

Motivation This problem comes from the situation where I have a non-random sample of individuals for which $p$ variables are measured. The target is to extract a subset of individuals which would be ...
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Confusion about range of integration for density function

Consider the joint density function: $$f(x,y) = \begin{cases} 2 & & \text{for } 0 \leq x \leq1 \text{ and } 0 \leq y \leq 1-x, \\[6pt] 0 & & \text{otherwise}. \end{cases}$$ From ...
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Can someone explain what the difference is in the p value of the summary output and emmeans (pairwise, least square means)?

I have a model: m_ramet<-glmer(ramets_net ~ salt * typhagroup + ramets_start + (1|site),data=morphdata, family="poisson"(link="log")) and am ...
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Where is my mistake in this definition of Bayes Factor?

From "The Bayesian Choice" by Christian P. Robert. The definition of the Bayes factor is given to be the ratio of the posterior probabilities of the null and the alternative hypothesis over the ratio ...
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Marginal Log Likelihood of Count Data to Simplify MLE

My question concerns the calculation of a marginal likelihood given some priors for my underlying exponential mixture distribution. Background My data is the (orderd) set of integers $\{N_\ell\}$. ...
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How to specify uniform distribution with same properties as normal distribution?

What I mean is, is it possible to specify a uniform random variable $U$ with random parameters $a,b$, where $a=-b$, and are generated from some other distribution, such that the marginal pdf of $U(a,b)...
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Marginal Distribution of Exponential Mixture Model

I am currently trying to marginalize over the scale parameter in a mixture distribution of exponential pdfs, but I do not trust my result. Let me show you my steps: Probability Density Function The ...
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174 views

marginal posterior distribution

Given $n$ normally distributed observations $f(X|\sigma_X)=\mathcal{N}(\mu_X, \sigma_X^2)$ and assuming a uniform prior on $\log(\sigma_X)$ and known $\mu_x$, I'm trying to find marginal posterior ...
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Joint densities and conditional independece

Let us assume the joint density $p(x,y,z)$ is factorized as $p(y)p(z|y)p(x|z)$. Hence, $x \perp y|z$. Now, the posterior distribution of z is: $p(z|x,y)=\frac{p(x,y,z)}{p(x,y)}$, where $p(x,y)=\int p(...
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marginalization of joint distributions

I am trying to understand the following sentence, section 2.2, in this paper: "...it is required that the joint mode $p(x,z,a)$ gives back the original $p(x,z)$ under marginalization over $a$, thus $...
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MCMC combined with numerical integration towards more efficient Bayesian inference

I am quite new to Bayesian statistics so the question can be a bit naive. My question is on how to deal with a model with individual coefficients. Simple versions of a task and a model I deal with is ...
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Prove that a multivariate density is valid

My question is quite simple. How to prove that a function $f(x_1,x_2,..., x_n)$ is a valid density? I mean, aside of the fact that must integrate to 1, and it must be positive, do i have to prove ...
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Multivariate distribution with specific (multivariate) marginal distributions

Let suppose I have a 6-variate random variable $\mathbf{x}=(x_1,x_2,x_3,x_4,x_5,x_6)$. What I want is to define a multivariate distribution for $\mathbf{x}$ with some specific multivariate marginals. ...
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About calculation of marginalizing (Bishop's book)

I'd like to ask a simple question, but I don't know how to solve it. Basically, it is from Bishop's book pattern recognition. The following figure is from his book. In chapter 8, to show that a and b ...
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MLE: Marginal vs Full Likelihood

Suppose I have a statistical model with parameters $\boldsymbol{\theta}=\{\theta_1,\theta_2,\dots,\theta_n\}$ of which only a single parameter, say $\theta_1$, is of interest to me. Suppose also that ...
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Scale mixture of normals; t-distribution; proving marginal distribution equality [duplicate]

This is a review-type homework question that I'm very much stuck on. Basically, these are the assumptions: $\theta|\lambda = N(\mu,\frac{\sigma^2}{\lambda})$ $\lambda = \rm{Gamma}(\nu/2,\nu/2)$ $\...
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Is it possible to derive joint probabilities from marginals with assumptions about the conditionals?

I understand the title is too generic. I tried to look for similar questions and although there were a few that were seemingly about the same issue, either they provided answers in the negative or had ...
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symmetric marginal but asymmetric joint distribution contours [duplicate]

Let us say we have two continuous random variables, $X$ and $Y$ such that their pdfs $f(x)= f(-x)$ and $g(y)= g(-y)$ for all $x$ and $y$. In other words, $X$ and $Y$ have symmetric distributions ...
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Fréchet Hoeffding bounds for symmetric random variables

(Edited to clarify the question). The Hakan & Demirtas (2012 doi: 10.1198/tast.2011.10090) approach to approximating Pearson correlation bounds uses the concept of the Fréchet-Hoeffding bounds by ...