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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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Changing the limits of integration when computing for marginal density [duplicate]

My question is based on this post. The question starts with $X \sim U(a, b)$ and $Y \sim U(a, X)$, and the answer computes the marginal density of $Y$. \begin{align} f(y) = \int_{-\infty}^{\infty} f(...
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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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1answer
142 views

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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24 views

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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15 views

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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1answer
33 views

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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1answer
51 views

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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1answer
70 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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1answer
19 views

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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26 views

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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2answers
66 views

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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1answer
25 views

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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52 views

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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1answer
36 views

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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2answers
86 views

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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2answers
106 views

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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1answer
74 views

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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1answer
218 views

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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1answer
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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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48 views

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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38 views

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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1answer
326 views

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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1answer
82 views

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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1answer
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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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1answer
48 views

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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1answer
49 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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31 views

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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0answers
50 views

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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1answer
57 views

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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0answers
40 views

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 ...
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0answers
28 views

Extracting Marginal Distribution by Monte Carlo [duplicate]

Suppose we know the distribution of a variable $x$ is $g(X)$, and we also know the conditional distribution of a variable $y$ given $x$ is $f(y|x)$. It has been shown that (such as in this book, p. 52 ...
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Why does test on Pearson correlation require bivariate normality?

For a pair of random variables $X$ and $Y$, we can compute their Pearson correlation coefficient $r$ and conduct hypothesis testing on the null hypothesis $H_{0}:r=0$ with the $t$ statistic $t=r\sqrt{...
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28 views

Joint density region of integration depends on a random variable

Suppose there is a joint density for $e_1$ and $e_2$, $f(e_1, e_2)$. In the double integral $\int \int f(e_1, e_2) de_1 de_2$, if the outer integral has a function of $e_1$ and $e_2$ in the region of ...
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0answers
42 views

Dirichlet disatribution: Connection between marginal and stick breaking distribution

Let suppose to have a probability vector $\boldsymbol{\pi} = (\pi_1,\pi_1,\dots , \pi_K)$, where by definition $\pi_K = 1-\sum_{j=1}^{K-1} \pi_j$, Dirichlet distributed with parameters $(\alpha_1,\...
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0answers
104 views

Understanding Partial Dependence for Gradient Boosted Regression trees

I am looking at the tutorial for partial dependence plots in Python. No equation is given in the tutorial or in the documentation. The documentation of the R function gives the formula I expected: $$...
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1answer
43 views

Solving a marginalization integral involving exponential distributions

I'm trying to solve a marginalization integral \begin{equation} \int p(y,w) dw \end{equation} in order to compute the density $p(y)$. I assumed the following model: \begin{equation} y = (u+w)^2 + v \...
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1answer
105 views

Does marginalization of some of the latent variables improve convergence in EM?

Given a likelihood to maximize $$ \log p(x | \theta) $$ Imagine that, in order to apply EM, we can augment the model with one or two latent variables. In that case, we can derive two lower bounds: $...
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Marginal density of bivariate density that is a circle with a hole

I have the following density function: $f_{(x,y)}(t,s) = \frac{1}{3\pi}$ for $1\le(t-2)^2+s^2\le4$ and else $f_{(x,y)}(t,s) =0$. I need to find $f_y(y) = \int{f(x)dx}$, but I having trouble to find ...
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0answers
145 views

Constructing a joint distribution from pairwise bivariate marginal distributions?

It's fairly well-known that given univariate distribution functions $F_X, F_Y, F_Z$, one can construct the joint distribution $F_{(X, Y, Z)}(x, y, z) = C(F_{X}(x), F_{Y}(y), F_{Z}(z))$, where $C$ is ...
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1answer
19 views

What does marginalising out means?

I am reading an article and it says the following and I quote: "In particular, if we define SNP-heritability h as the proportion of variance explained by the SNPs included in the model after ...
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19 views

sequential versus marginal in nlme()

I'm analyzing experimental data with two factors: group (2 levels) and condition (5 levels). Group is between subjects and condition is a repeated measure variable. The design is completely balanced ...
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62 views

Parametric family problems

I came across such a problem that I cannot solve: Let $\mathcal{P} = \{\mathbb{P}_\theta : \theta \in \mathbb{R}\}$ be a parametric family over $\{0,1\} \times \mathbb{R}$ defined in the following ...
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1answer
343 views

Marginalizing conditional probabilities conditioned on multiple variables

I am wondering if the following equality of marginalized conditional probabilities holds $\sum_{x} p(x \vert y) = \sum_{x} p(x \vert y, z) = \sum_{x} p(x \vert y, z, w) = \sum_{x} p(x \vert y, z, w, ...