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

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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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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 ...
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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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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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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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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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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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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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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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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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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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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, ...
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Marginal Likelihood as probability distribution

I understand that Likelihood is not a probability distribution but what about Marginal likelihood ? Suppose we are given a set of data points $\pmb{X}$ such that each data point $x_{i} \sim p(x_{i}|\...
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Can I use interaction/ quadratic variables in Fractional (logit/probit) response regression?

Can I use interaction/ quadratic variables in Fractional (logit/probit) response regression? I'm able to obtain log-odds ratio results for the quadratic variables (e.g. c.X1##c.X1) when running the ...
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Constraints on marginal distribution given joint distribution definition

I am trying to understand a part of the paper 'From optimal transport to generative modelling: the VEGAN cookbook' by Bousquet, Gelly, Tolstikhin, Simon-Gabriel and Schölkopf. There is a one-liner in ...
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166 views

Finding the marginal densities of uniformly distributed X, Y

If (X, Y) is uniformly distributed over the region defined by $$ 0 ≤ y ≤ 1−x^{2} $$ and $$ −1 ≤ x ≤1 $$, what would the marginal densities of X and Y be? I have determined that, for X: $$ f_{x}(x) = \...
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Is it necessary to report estimate of variance for a marginal likelihood estimation?

There are several simulation based estimators for estimating marginal likelihood of data - Importance Sampling estimator, Newton Raftery estimator, Chib's method, Gelfand-Dey estimator etc. But while ...
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Tuning density in Gelfand-Dey estimator (Reciprocal Importance Sampling) of Marginal Likelihood

If y denotes the data and (t,L) denotes set of parameters, then the marginal likelihood is Here, is a proper prior, f(y|t,L) denotes the (conditional) likelihood, m(y) is used to denote the marginal ...
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Marginal likelihood derivation for normal likelihood and prior

For a normal likelihood $$ P(\mathbf{y}|\mathbf{b}) = \mathcal{N}(\mathbf{Gb}, \mathbf{\Sigma}_y) $$ and a normal prior $$ P(\mathbf{b}) = \mathcal{N}(\mathbf{\mu}_p, \mathbf{\Sigma}_p) $$ I'm trying ...
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Find the joint distribution of two independent random variables

Two random variables such as $X_{1}, X_{2},...,X_{n}$ be iid's has pdf $\theta x^{\theta-1}$ where $0<x<1$ and $Y_{1}, Y_{2},...,Y_{n}$ be iid discrete random variables have power series ...