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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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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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Finding a marginal distribution from other overlapping marginal distributions

The setup: Say you are interested in the marginal distribution $P(X_1, X_2, X_3)$ from the joint distribution $P(X_1, X_2, X_3,X_4)$, but you do not have access to the joint distribution. You do, ...
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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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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 ...
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Avoiding underflow errors when marginalizing over a nuisance parameter in Bayesian inference

I was reading this question about how to marginalize over nuisance parameters in Bayesian inference, and the concern I have is how to deal with underflow errors. If we are interested only in the ...
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Highest Posterior Density (HPD) region of the marginals vs. of the joint distribution

In a Bayesian context, to analyse the posterior distribution, one can define the Highest Posterior Density (HPD) region or interval as $$\{\theta; \pi(\theta \mid x) \geq k\} $$ in both unidimensional ...
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The distribution of the initial point of an AR process

Consider a stochastic process $\{X_t, t = 1, 2, \ldots\}$ following the model $$X_t = \alpha X_{t-1} + e_t,$$ where $e_t \thicksim f$. Can I say that the distribution of the initial point, $X_1$, ...
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What is the differences between estimating the margins and transforming them using cumulative distribution function

In copula models, the estimation of copula parameters is based on the pseudo-observations of the original data. As I understand, we can transform the margins using the cumulative distribution function ...
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Marginal medians of the Dirichlet distribution

I am working with a 3 dimensional Dirichlet distribution with parameters $\alpha_1,\alpha_2,\alpha_3>0$. I have been trying to figure out a useful 'median' concept for this distribution. The vector ...
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How can marginalizing over intermediate variable give arbitrarily complex distribution

In this web page has the following statement regarding an inference model $$ q(z_2,z_1|x) = q(z_2|z_1) q(z_1|x) $$ "Although we are still sticking to Gaussians for all of the factorized ...
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marginal distribution of multivariate normal

I am approximating a probability distribution over n RVs by n factors following a bivariate conditional distribution. For instance for 4 variables I could factorize p as: $$p(x_1,x_2,x_3,x_4) = p(x_1)...
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How does probabilistic ML “handle uncertainty”?

I have heard professors and others say that probabilistic machine learning is useful because it can model or handle uncertainty. I'm not sure what is meant by this. To give an authoritative source, ...
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Marginal density of $X_1$ given that $X_1 + X_2 = d$ where $X_1$ and $X_2$ are iid Weibull?

In their tutorial (page 23) on heavy-tailed distributions, Nair et al. present the following graph (taken from a pre-publication chapter from a book by the same authors): Pictured are the marginal ...
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Mass point delta and it's mathematical interpretation

Considering a spike-and-slab prior of the form $$w\sim\pi\mathcal{N}(0,\alpha^{-1})+(1-\pi)\delta_0$$ where $\delta_0$ is a point mass at zero, if we would like to integrate over w such that $$I=\...
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55 views

multinomial distribution aggregation property

Suppose we have multinomial distribution in which we have 4 categories, and each one is associated with a probability of being selected, say $\theta_i$, $i=1,..,4$. And I know for sure that $\...
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Copulas - what marginals can I use? (theory)

For my research I am using various copulas and I fit different marginal distributions to my data. I've studied the topic of inter-variable dependency quite a bit, however, I do not recall the ...
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computing marginal dependence

I have three random variables $x$, $y$. Let $z$ both $y$ and $z$ depend on $x$ through known distributions $Pr(y|x)$, $Pr(z|x)$ but be otherwise independent: $y \perp z$. The distribution of $x$ is ...
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Creating a joint density from two asymmetric uniform marginals

Two RV \begin{split} X_1 & \overset{{i.i.d.}}{\sim}\mathcal{U}[0,1] \\ X_2 &\overset{{i.i.d.}}{\sim}\mathcal{U}[0.3,2.08] \end{split} I need to create a joint pdf from these two. Is copula ...
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Maximizing likelihood for noise-free Gaussian process regression

In "Machine Learning: A Probabilistic Perspective" the maximum marginal likelihood optimization for the kernel hyperparameters is explained for the noisy observation case. I am dealing with a noise-...
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What is the difference between using a standard logistic regression model versus a marginal model?

I have been analyzing a longitudinal dataset and came across a peculiar finding. I have been using a marginal logistic regression GEE to analyze the regression coefficients, standard errors, Z-scores ...
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Marginal prior derivation in hierarchical Bayesian model

I am working on a model that is closely related to the normal gamma shrinkage prior setup discussed in Griffin & Brown (2010). Suppose we want to draw $P$ parameters $\beta_p$ with $p=1,...,P$. ...
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Finding credible intervals for hyperparameters in Bayesian inference

I'm trying to use Bayesian inference to fit and interpret a linear model of the following form: $$ y=X\beta + \epsilon \hspace{1cm} \text{where } \epsilon_i \sim \mathcal{N}(0,\sigma^2)$$ The ...