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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Let random variables X and Y have the following joint density [closed]
Let random variables X and Y have the following joint density:
fX,Y (x, y) = e^−y for 0<x<y; 0 otherwise;
Find marginal density of X and Y, conditional density of X|(Y = y) and show that (Y − X)|...
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Finding the non-zero region of a marginal
Basic question but: what happens to the region on which a pdf is non-zero when a bivariate is integrated to get a marginal? The example I'm working on (course problem booklet for a mathematics BSc ...
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The PDF of the Data Given (Marginal Likelihood) the Likelihood and the Prior of a Normal Distribution with Prior on the Mean
Given a model where $ x_i | \mu \sim \mathcal{N} ( \mu, \sigma^2 ) $ where $ \mu \sim \mathcal{N} ( \mu_0, \sigma_0^2 ) $, is there a closed form formula for the PDF of $ x_i $? Namely, what's $ p (...
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Integration in R: "longer object length is not a multiple of shorter object length" Error [migrated]
I am trying to use R to calculate the marginal likelihood of a set of data (likelihood_N) with respect to the parameter $v_0$. To do this, I created the following ...
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Marginal posterior density for Box-Cox transformation
I am trying to solve Chapter 7,Problem 5 from Bayesian Data Analysis (3rd edition) by Gelman et al.
Part (c) asks us to find the marginal posterior density $p(\phi|y)$ with the prior form given in ...
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Infer $p(x|y)$ given $p(y|x)$ and $p(y)$
Given $p(y)$ and $p(y|x)$, how can we infer $p(x|y)$?
Or to what extent can we know about $p(x, y)$?
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Marginal Distribution of a Stochastic Block Model?
Let us say I have a Stochastic Block Model i.e. a random vector $X^{(n)} = [x_1,...,x_n]$ where $P(x_i = c) = P_{c}$, where $c \in \{1,...,u\}$, and a random simple undirected graph represented by $Y^{...
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Marginal posterior distribution of error variances
I have been working on Bayesian statistics recently and have came across the term called Marginal distribution of error variances. Though I understand what is a marginal distribution and that an error ...
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Hierarchical model for uniform random variable
I am thinking about the following model:
$$
\theta \sim \mathcal{U}[c- \epsilon, c+\epsilon],\\
x \mid \theta\sim \mathcal{U}[\theta - \epsilon, \theta + \epsilon].
$$
I want to derive the marginal ...
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Expectation of a joint distribution vs product of marginal expectations?
I got into a conversation with a coworker, he was doing napkin math and showed that average purchase value and average conversion rate, together, give us the expected revenue. And my response was- not ...
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Integrating out variables in a Gaussian copula density
Consider a multivariate continuous distribution with a Gaussian copula, i.e. we can write its PDF as
$$
p(x)= \biggl(\prod_{j=1}^D p_j(x_j)\biggr) c\bigl(F_1 (x_1), \dots , F_D(x_D)\bigr) \enspace,
$...
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Proof for a seemingly simple property for fully-connected coloured random graphs?
I have a probability distribution defined over a set of fully-connected simple graphs depending on their coloring. Let us have a fully connected graph with $N$ nodes, a node may have a color $i$ ...
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marginal and conditional distributions
Can someone help me, please?
Let X and Y be two random variables having the following marginal and conditional distributions.
𝑌|𝜇 ~ 𝑃𝑜𝑖(𝜇) 𝜇 ~ 𝐺𝑎𝑚𝑚𝑎(𝛼, 𝛽)
I want to obtain the ...
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"Completely" Marginalizing Out a Variable from a Probability Distribution
Suppose you have a multivariate probability distribution function for 4 variables X1, X2, X3, X4 : P(X1, X2, X3, X4)
Normally, you can write P(X1|X2 = x2, X3 = x3, X4 = x4) : Then, you can find also ...
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Is a draw from the posterior always the same as a draw from the prior?
I'm reading Bayesian data analysis. On page 155, the authors state:
Each of the [...] parameters were assigned independent Beta(2, 2) prior distributions. ... If the model were true, we would expect ...
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Use standard deviations to find for marginal distribution for each variable
I have a data TVbo described by:
The main purpose was to assess 12 different TV sets (products) specified by the two attributes
Picture and TVset. 15 different response variables (characteristics of ...
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why do some use ν=n instead of ν=n-1 in the t distribution
I have always accepted that ν=n-1, but then I come across multiple authors showing a derivation of the t distribution by marginalizing out the variance from the normal-inverse-gamma distribution, or ...
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Simulation in R to check graphically that marginal distributions are correct
The distribution on $R^2$ with joint density $h$ with respect to the Lebesgue measure is:
$$h(x,y)=\frac{3}{2}y 1_{A}(x,y), \ \ A=\{(x,y) \in R^2|0<y, x^2+y^2<1\}.$$
Then I have found the ...
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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 [duplicate]
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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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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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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