Questions tagged [marginal-distribution]

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

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Law of total probability for random variables with Y < X

Could someone explain to me why the following equation holds? It is related to the law of total probability , but I don't get it. I'm confused because it's two random variables on the right side and ...
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Marginal likelihood: Why is it difficult to compute in this case?

I have been reading up a bit on generative models particularly trying to understand the math behind VAE. While looking at a talk online, the speaker mentions the following definition of marginal ...
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Question re: Marginal pmf for Joint Discrete Variables (Textbook Exercise)

Given a discrete joint distribution with pdf $ f(x, y) = p^y (1 - p)^2 $ for $ p \in (0, 1) $ and $ x, y \in \mathbb{N}, x \leq y $ (here I include 0 in the natural numbers), find the marginal ...
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Using Monte Carlo to sample from marginal distribution

I am defining a model on a vector, $T$, of size $n$, wherein each element $t_i \in T$ is independent and either $0$ or $1$. This model depends on 3 other parameters, $q$ (also a vector of size $n$), $\...
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Cumulative incidence of X

Suppose the joint survival function of the latent failure times for two competing risks, $X$ and $Y$, is $S(x,y)=(1-x)(1-y)(1+0.5xy)$, $0<x<1$, $0<y<1$. Find the cumulative incidence ...
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Is it always possible to find a joint distribution $p(x,y)$ consistent with the results of both labs?

I am reading "Bayesian Reasoning And Machine Learning" and doing exercise 4.8 and would like to check if the following reasoning is correct. Two research labs work independently on the ...
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Is there a difference between marginal likelihood and likelihood of a marginal distribution?

Assume that we have $n$ i.i.d. samples $x_i \sim f(.|\theta )$, where $\theta$ is a parameter. Furthermore, the parameter is also distributed as $\theta \sim \mathcal{g}(.|\alpha) $. Now let's say ...
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Find marginal distribution of X (Bayesian setting)

$X|\theta$ follows $N(\theta,w)$ and $\theta$ follows $N(\mu,\sigma^2)$ Both follow a normal distribution but with different mean and variance. I assume it is a Bayesian setting. How to find the ...
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Are the mean of a joint distribution and its marginals equal?

I know that for a multivariate normal distribution the mean of the parameters is the same as the mean of the marginal distributions for the respective parameters. But is this always the case?
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Normalizing constant in Bayes' Theorem

As we know, Bayes' Theorem is given by: $$P(\theta\vert{D})=\frac{P(\theta)P(D\vert\theta)}{P(\theta)P(D\vert\theta)+P(\neg\theta)P(E\vert\neg\theta)}$$ where $\theta$ is the hypothesis and D is the ...
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Is outer product of marginal distribution the "best" mean-field approximation for a joint distribution?

I am certain this has been asked somewhere else, if that's the case, link me and close the thread. I am studying variational inference and mean-field approximation. All the explanations I come across ...
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Can a (possibly infinite) mixture of Gaussians be Gaussian?

Suppose we define a (possibly infinite) mixture of zero-mean Gaussians: $$p(x) = \int_{\mathbb{R}^+} N(x; 0, \sigma^2)\ \pi(\sigma)\,\text d\sigma,$$ where $\pi$ defines the mixture components. ...
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Sampling from marginal densities using the joint

I have seen this approach link in different places. Hence, I suppose that the following method is correct. Let $f_{X,Y,Z}(x,y,z)=f_{X|Y,Z}(x|y,z)f_Y(y)f_Z(z)$ and I can easily sample from these 3 ...
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quantile surface of a mulitvariate distribution made of multiplication of marginal distributions assuming independence

How to perform quantile regression in a more elegant fashion? As discussed above, quantSheets() can only deal with one explanatory variable for computing quantile ...
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How can the marginal distribution be derived from conjugate Gaussians?

In An Introduction to Empirical Bayes Data Analysis by George Casella (1985), it is given that \begin{align} x|\theta &\sim N(\theta,\sigma^2) \\ \theta &\sim N(\mu,\tau^2) \end{align} and ...
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How can we assume the models are exhaustive in Bayesian Model Averaging?

Bayesian model averaging is justified using the law of total probability which requires the the set of models that we average over to be exhaustive. Shouldn’t we prove that the set of models are ...
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Exponentiated Weibull-logarithmic Distribution

I'm trying to deduce the marginal cdf of $Y$ in Exponentiated Weibull-logarithmic Distribution from this paper: Exponentiated Weibull-logarithmic Distribution: Model, Properties and Applications In ...
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Noob question about bayes rule denominator estimation

A known problem of Bayes rule is the intractability of the estimation of $p(D)$ given a multiparametric problem, since $p(D)$ is found by marginalizing the joint probability $p(D, \theta_{1..n})$ over ...
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Approximating joint distribution from marginals and additional information

Consider a population generation question where we are trying to generate couples that conform to a local areas demographics. We know the age distribution for Partner 1, $x_1\sim D_1$, and for ...
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Estimate a marginal (general) likelihood of best success time

In the figure below, I estimated two nonparametric pdf's: (x-axis is time in 24hour format, range between 8 to 20) green line: clock time of success phone calls (eg. at 4pm I called the customer, and ...
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How does this transformation work?

I encountered this expression; How do we get $p(x|A)$ from $p_x(x[n]|A)$? I thought we just sum all discrete x[n] values, but in another example there was this one which got me confused. $p(x[n]|\...
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Combining marginal bivariate probability distributions into a single multivariate one

Consider a set of 3 correlated random variables $X$, $Y$ and $Z$. I have calculated bivariate marginal distributions over any pairs of these random variables. That is, I know probability density ...
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Marginal distributions of the Indian Buffet Process

In the construction of the Indian Buffet Process we have that customer $n_1$ chooses $\mbox{Poisson}\left(\frac{\alpha}{1}\right)$ dishes, $n_2$ chooses $\mbox{Poisson}\left(\frac{\alpha}{2}\right) \...
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Finding the marginal distribution of log-normal random variable whose mean is dependent on a Gaussian random variable

My goal is to be able to integrate out the observation error of $\hat{X}$ in the set-up below, in order to compute the likelihood of $\frac{X}{\hat{X}}$ over all possible values of $\hat{X}$ : The ...
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Intergrating product of multivariate normal and univariate normal to find marginal density [duplicate]

Suppose, $y_i|u_i\sim MN(X_i(t_i), \sigma_e^2I_{m_i})$ and $U\sim MN(0,I_p)$. Now how to find the marginal distribution of $f(y_i)=\int f(y_i|u_i)f(u_i)du_i$? Since one is multivariate normal and ...
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Integrating product of multivariate normal and univariate normal to find marginal

Suppose, $y_i|u_i\sim MN(X_i(t_i), \sigma_e^2I_{m_i})$ and $u_i\sim N(0,1)$. Now how to find the marginal distribution of $f(y_i)=\int f(y_i|u_i)f(u_i)du_i$? Since one is multivariate normal and other ...
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Generating an analytical copula for an example problem

I am currently doing research that requires me to understand dependence modeling. As a first step, I am reading An Introduction to Copulas. I am, stuck on the first example problem which I have re-...
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Marginal and joint distributions of normal and student's

Is there an equivalence between two normal (student's) distributions. in the sense that, if my two variables (X,Y) are normal (student's) then their joint distribution is also normal (student's). I ...
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Can we calculate the marginal distribution via Importance Sampling?

I just have a quick question about calculating marginal distributions via importance sampling. Let's say I have a function, $p(x,y)$, which is two-dimensional. And, I wish to integrate out one ...
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Marginal distribution of uniform distribution over sphere

Let $(x_1,…,x_n)$ be a random vector uniformly distributed on the $n$-dimensional unit sphere. Is there a closed form solution for the joint distribution of $P(x_1, x_2)$?
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Limits of integrals for marginal pdf and expected values [duplicate]

The joint probability density function of independent random variables $X$ and $Y$ is $\frac{1}{34}$ on the region$0<x<6, 0<y<6, x+y<10.$ I am trying to find $E[X]$ however I am stuck ...
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How to derive marginal likelihood equation for linear regression as product of two Gaussians?

I understand that marginal-likelihood can be derived as answered here. Quoting the same proof from MATHEMATICS FOR MACHINE LEARNING book (9.3.5) Page 312, The same book mentions that we can derive ...
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Marginal distributions given the distribution of range

I'm working with an upper diagonal distribution whose distance from the diagonal is Lomax Pareto (Type II) distribution. The distance of a point from the diagonal line y = x is $\frac{\sqrt{(x_0-y_0)^...
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Help me understand this line of proof which concerns a marginal expectations in the presence of independent variables

The following comes From Martin Wainwright's book on High-Dimensional Statistics, page 41 on Lipschitz functions of Gaussian variables. It first begins by the following Lemma: Suppose $f:\mathbb{R}^n\...
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Inferring properties of the sum of R.V.s from the copula

This is not a completely well defined question, so even help making it coherent will be useful. Setting: Suppose I know the marginal distributions of random variables describing the expected losses ...
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Confusion about bounds of integrals when calculating marginal density

I've already read this question on StatsSE, but the answer is not satisfying to me. I would like help "fixing" my lecture notes, because I believe they are unclear/inaccurate, or even ...
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Is the sample log marginal likelihood divided by its dimensionality n constant as n increases?

Hi everyone and thank you all in advance! I am a physicist working on Multivariate Gaussian distributions (not with a really strong theoretical math background). Let's have a sample drawn from a ...
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Marginal likelihood of bivariate Gaussian model

I assume the following model for a sample $y_1 \in \mathbb{R}^2$ of size $1$ with bivariate Gaussian likelihood and independent bivariate Gaussian and inverse-Wishart prior for the mean and variance ...
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How to jointly model $N$ groups where data in each group is i.i.d. Normal and infer the posterior distribution?

I am given the following data of income scores of individuals from $N$ groups: $$(\textbf{x}_1, \textbf{x}_2 \ldots \textbf{x}_N),$$ where $$\textbf{x}_j = (x_j^1, x_j^2 \ldots x_j^{N_j}),\quad j = 1, ...
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Marginal Distributions obtained by restricting a 2D Gaussian to a circle

Suppose I have a 2D Gaussian $$ f(x, y) = \frac{1}{2\pi\,\sqrt{\text{det}(\Sigma)}}\exp\left\{-\frac{1}{2}(\boldsymbol{x}- \boldsymbol{\mu})^\top \Sigma^{-1} (\boldsymbol{x}- \boldsymbol{\mu})\right\} ...
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completing a square

If I have a density function of the form $p(x) \propto \exp(−q(x)/2)$ where $q(x)$ has the following quadratic function $$q(x)=x^Tx+y^Ty-[x^TA+y^TB][A^TA+B^TB+\beta\mathbb{I}]^{-1}[A^Tx+B^Ty]$$ where $...
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Transform Copula Marginals to come from Mixture Distribution

I am trying to create a joint distribution that has a specific copula (e.g. Clayton) and whose marginals come from a mixture distribution (e.g. the mixture of two Gaussian distributions). My idea was ...
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Marginal distributions of two linear transformations of two correlated Gaussian (Normal) distributions

Considering this entry the distribution of the sum of non i.i.d. gaussian variates is also gaussian. $$ \begin{align*} V = aX + bY &\sim N(a\mu_X + b\mu_Y,\; a^2\sigma_X^2 + b^2\sigma_Y^2 + 2ab\...
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Calculating marginal distribution of Markov process

i am studying markov processes and we have an example of a VAR process. i am trying to understand how to look at the marginal distribution so i can find a gaussian distribution for $Y_t$ which equates ...
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How to interpret marginal probability of a dataset?

I was going through a survey paper on transfer learning available at https://arxiv.org/pdf/1911.02685.pdf. Under section 3.2, (see attachment), authors have defined the domain as being composed of ...
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Multivariate Normal Distribution: Divide each random variable by its standard deviation

If $X$~$Normal(\mu,\Sigma)$, and I divide each random variable in $X$ (the marginals) by its standard deviation, what will happen to the covariance matrix $\Sigma$?
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Constructing a joint distribution from pairwise marginals

Consider a set of random variables $\{X_i\}$ with joint pdf $f(x_1 ... x_n)$. Given the marginal pdfs $f_i(x_i)$, we can construct a joint distribution $$g(x_1 ... x_n) = \prod_i f_i(x_i)$$ which has ...
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Do the following equations for the conditional expectations hold under the given assumptions?

Let $(\Omega,F,\mathbb{F}=(F_n)_{n\in\mathbb{N_0}},P)$ be a filtered probability space and $(Z_n)_{n\in\mathbb{N_0}}$ be a sequence of integrable iid random variables. Denote $F_n=\sigma(Z_o,\dots,Z_n)...
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Marginal likelihood of a Gaussian Process?

I've been studying Gaussian processes (GP), this resource implements GP from scratch and has been very useful in visualizing what's happening. So far all has made sense to me except for the below ...
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Why do elliptical copula densities contain $x_1$ and $x_2$, but Archimedean copula densities contain $u_1$ and $u_2$?

$$c\left(u_{1}, u_{2}\right)=\frac{1}{\sqrt{1-\rho_{12}^{2}}} \exp \left\{-\frac{\rho_{12}^{2}\left(x_{1}^{2}+x_{2}^{2}\right)-2 \rho_{12} x_{1} x_{2}}{2\left(1-\rho_{12}^{2}\right)}\right\}$$ is the ...

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