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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Marginal distribution of an autoregressive process of order one AR(1)

I'm reading "Econometric Modelling with Time Series" by V. L. Martin, A. S. Hurn and D. Harris ( https://www.researchgate.net/file.PostFileLoader.html?id=56bccdaa6225ff0de28b45a6&...
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Is the posterior maximum always the same as the marginal's?

When I see plots of the conditional probabilities, marginals and joint distributions together they are mostly plot using Gaussians. It is not clear to me if this applies to every other distribution. ...
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2 votes
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For log-concave densities, are joint and marginal modes consistent?

Suppose I have a probability density function $\pi(x_1, \ldots, x_n)$, which is the density of a vector-valued random variable $X$ in $\mathbb{R}^n$. Assume that $\pi$ is strongly log-concave, i.e., ...
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1 answer
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Creating multivariate distribution using marginal: can I use a copula?

I am modeling a multivariate distribution, where I already have the distributions of the marginals. Let's call the marginals, $f_{i}(x)$ for every $i$ marginal distribution, where $i=1,2,3$. We ...
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2 votes
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Calculate expected values E(x) & E(y) & variance of x & y of joint PDF, which was previously transformed from Polar to Cartesian

Given two independently uniform distributed random variables angle $\theta \in [0,2\pi]$ and radius $r \in [0,1]$. I obtain for the joint density function with polar coordinates: $$ f_{r,\theta}(r,\...
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When do marginal/conditionally normals imply joint normal?

I know that marginal normals do not imply joint normal, as some examples in here gives. However, but I don't know of any theorems that talk about conditions under which marginal normals can imply ...
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Compute Mutual Information from a fixed dataset

The goal is to compute the mutual information between two variables from the fixed dataset. $$I(X;Z)= E_{P_{XZ}}[T(X,Z)]-log(E_{P_X\otimes P_Z}[e^{T(X,Z)}])$$ This equation is taken from this paper. ...
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Sum of Multivariate Distribution with Normal Copula

When summing multivariate distributions that uses a normal copula, is it true that the normal copula part of the distribution remains unchanged? The R code below simulates from a two-dimensional ...
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Marginal mean distribution with Gaussian Process

The marginal posterior over the mean for normally distributed data with a known variance $\sigma^2$ is (from here) normally distributed, with parameters: $$ \mu' = \frac1{\frac1{\sigma^2_0} + \frac{n}{...
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Marginal Likelihood in Gaussian Process Regression with derivative observations

I have a question regarding the calculation of the marginal likelihood within Gaussian Process Regression when derivative observations are incorporated. If $y=(y_1,\dots,y_n)^T$ are noisy observations ...
3 votes
1 answer
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Latent variable model: why is the marginal probability of data under the model intractable?

I'm reading about Variational auto-encoders, and in section 1.8 they describe a latent variable model $p_\theta(x,z)$ where x are observed and z are hidden variables. They say that the marginal ...
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Marginal density of dirichlet distribution

I'm studying BRML. In this book, a Dirichlet distribution is defined as $$ p(\alpha | u) = \frac{\Gamma(\sum_{q=1}^Q u_q)}{\prod_{q=1}^Q \Gamma(u_q)} \delta_0 \left( \sum_{q=1}^{Q} \alpha_q - 1 \right)...
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If X and Y have the same marginal distribution, then do they have to be equally distributed?

I have found an exercise with two variables X&Y and this joint distribution table: X\Y -5 -4 -3 -2 -1 0 1 2 3 4 5 X 2 0 0 0 0 0 1/36 0 0 0 0 0 1/36 3 0 0 0 0 1/36 0 1/36 0 0 0 0 2/36 4 0 0 0 1/...
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Why does $E\left(y_{i t}\right)=a^{\prime}\left(\theta_{i t}\right)$? in the context of assuming some GEE marginal density?

In generalized estimating equations we have a glm-response variable. To establish notation, we let $Y_{i}=\left(y_{i 1}, \ldots, y_{i n_{i}}\right)^{\text {T }}$ be the $n_{i} \times 1$ vector of ...
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A question about the population marginal distribution when using survey raking method

Sorry if my question might seem naive, but I was wondering about the creation of the marginal distribution table of the target population of a conducted survey. The objective is the use of the raking ...
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Optimal sampling strategy when only marginal group sizes are known

I seems to me that this cannot be such an uncommon situation but I find I cannot come up with the answer myself and I also cannot come up with the correct search terms to find it, either here on Cross ...
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Derive and validate a probability using a causal diagram

Given a causal diagram and the conditional probabilities for every adjacent node, I want to calculate a specific probability in two ways - let's call them the "simple" and "alternate&...
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marginalisation with conditional probability

In this link, Marginalization with conditional probability is answered. Conditional probability with chain rule and marginalisation I am little bit confused about my another way of thinking and would ...
2 votes
0 answers
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Is it circular reasoning to compute the ELBO using MCMC?

Let's say we have a posterior distribution $q(\theta) = p(\theta \mid D, \mathcal{M})$ over parameters $\theta$ given data $D$ and a model $\mathcal{M}$. As is often the case, computing $q$ is hard, ...
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Construct a joint probability mass function from marginals

I want to construct a joint probability mass function from its marginals, under some constraints. I would like to understand if this is always possible. Below, I formalise this question in terms of ...
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generalization of univariate pdf under constraints

I'm looking for generalizations of a univariate probability distribution function. The pdf is $$ \varphi(x)=(2\sqrt{s}K_1(2\sqrt{s}))^{-1}e^{\frac{s}{\log x}}. $$ for $K_1$ a modified Bessel function, ...
3 votes
1 answer
369 views

Drawing from the joint distribution by drawing from the marginal and conditional distribution

How can we prove that a draw $(x,y)$ from the joint probability distribution of two random variables $X$ and $Y$ can be obtained by first generating a draw $x$ from the marginal probability ...
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Computational Complexity of Marginalization

Suppose you have a joint probability distribution with M variables, each sampled from a set of cardinality N. Now, suppose you want to marginalize one of the variables. My guess right now about the ...
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Intuition about the relation between joint distribution, marginal distribution, and conditional distribution

The wording "intuition" might be a bit imprecise. I want to discuss how we visualize in our head going from one to another among the joint PDF, marginal PDF, and conditional PDF. To make the ...
1 vote
1 answer
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Expand conditional by marginalisation and drop terms from conditional

I have come across this conditional expansion a few times, and I can't seem to make sense of it. $$p(z|y) = \int{p(z|f)p(f|y)df}$$ I would go about it like this: \begin{align} \require{cancel} p(z|y) &...
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Margin Distribution N-dimensional random vector

I have this exercise on my textbook and I can't understand how I have to do it. I tried to compute the posterior but I don't understand if it's what the exercise requires. Thanks This is what I did ...
1 vote
1 answer
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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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2 answers
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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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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 ...
3 votes
1 answer
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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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2 votes
1 answer
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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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1 vote
0 answers
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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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1 answer
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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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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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2 votes
1 answer
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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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4 votes
3 answers
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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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1 answer
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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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1 answer
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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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1 answer
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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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2 votes
1 answer
219 views

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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2 votes
1 answer
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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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1 vote
1 answer
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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 ...
4 votes
1 answer
509 views

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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1 answer
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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 ...
1 vote
1 answer
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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 ...
0 votes
1 answer
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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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8 votes
1 answer
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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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