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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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Are the marginals of these two joint distributions the same? [closed]

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and let $X:(\Omega, \mathcal{A})\rightarrow (\mathcal{X}, \mathcal{F})$ and $Y:(\Omega, \mathcal{A})\rightarrow (\mathcal{Y}, \mathcal{G})...
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Gaussian linear model marginal likelihood under g-prior

Consider a Gaussian linear model with an $ n \times 1 $ outcome vector $ y $ and an $ n \times p $ matrix of centered predictors $ X $: $ y = \iota\alpha + X\beta + \varepsilon \quad \quad \varepsilon ...
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Marginal probability expressed as posterior expectation [closed]

I'm reading Probabilistic Machine Learning: Advanced Topics by Kevin Murphy and fail to see how, on page 339, plugging $g(\boldsymbol{\theta}) = p(\theta_1=\theta_1^*\vert\boldsymbol{\theta}_{2:D})$ ...
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How to Compute the Posterior Distribution of Covariance matrix in a Matrix Normal Model with Inverse Wishart Prior

I am working on a time series model involving Kalman filters and smoothing to estimate state variables $Y_i$. The part of model is structured as follows: $Y_1, \ldots, Y_n$ are iid. $Y_i \sim \mathcal{...
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How to choose what to integrate out or what to condition on for marginal distributions?

I am trying to work out the Bayesian posteriors of $\theta$, $\tau$ and the $\varepsilon$ in the following model: $$y(t) = \phi(t,\tau)\theta+v(t),$$ where $\{v(t)\}$ is an iid sequence of random ...
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Modelling the joint pmf of 2 correlated variables as p(x)*pmf(E(y|x))

Let x,y be 2 correlated counts. We want to model the joint pmf p(x,y). We know that p(x,y) = p(x)p(y|x) = p(y)(x|y). However, what happens when we don't know y|x, but we can estimate E(y|x)? Can't we ...
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MLE of marginal distribution for continuous random variable

Let $\mathcal{F}$ be a family of multivariate probability densities such that for a sufficiently large data sample, there always exists a unique MLE. Assume also that all marginal and conditional ...
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Given pairwise marginal distributions, what can we say about the full joint?

From reading previous posts, I understand that, if we have pairwise marginals, say $P(A,B)$, $P(B,C)$, and $P(C,A)$, this doesn't allow us to reconstruct the full joint $P(A,B,C)$. But does it allow ...
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How to obtain $p(x)$ given samples from $p(y|x)$ and $p(y)$?

Here, assume both $p(y\mid x)$ and $p(y)$ are too complicated to get closed forms, and we can only draw samples from them. Is there any way to estimate or draw samples from $p(x)$?
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Constructing Joint Multivariate Normal Distribution from Marginal Distribution (detrended data)

I have two n-length data vectors $X_1 \sim N(\mu_{1},\Sigma_{1})$ and $X_{2} \sim N(\mu_{2}, \Sigma_{2})$ which may or may not have a covariance. To see whether they do or not, I detrend them by ...
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Tractability of the E Step of the Expectation Maximization Algorithm

To optimize a probabilistic model with latent variables, I was trying to convince myself that there are situations where computing the marginalization $p(x)$ is intractable and computing the $Q(\theta ...
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MCMC seems very sensible to the evidence

currently starting to study bayesian ML, and specifically MCMC, in order to compute the posterior: $$ P(\theta|D) = \frac{P(D|\theta)P(\theta)}{P(D)} $$ Now, I see how the acceptance ratio makes sense ...
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Marginalising and taking the log of the likelihood

Say you have a likelihood function $L(X|\theta_1,\theta_2)$ and you marginalise out $\theta_1$ and then take the log of the marginalised likelihood function. Would this be the same as taking the log ...
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Mean of normal follows a T distribution

Suppose: $x \sim \mathcal{N}(x; \mu, \Sigma) \;\;\;$ st. $\;\;\; \mu \sim T_{v}(\mu; k, M)$ Where $T$ is the $t$-distribution with v degrees of freedom, location $k$, and shape $M$. Then, is there a ...
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Resulting univariate marginal distributions are not $t$ distributed - why? (S Wood Core Statistics)

In the Core Statistics by Simon Wood it says: "If we replace the random variables $Z_i\sim_\text{i.i.d.} N(0,1)$ with random variables $T_i \sim_\text{i.i.d.} t_k$ in the definition of a ...
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Copula to ensure one team wins and the other loses (Bernoulli margins)

Team $1$ has a historical win percentage of $p_1$. Team $2$ has a historical win percentage of $p_2$. The upcoming game features team $1$ against team $2$ and cannot end in a tie (one team wins, and ...
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Numerical Optimization of Marginal Likelihood that Explodes

I have a model with a marginal likelihood of the following form: $$\mathcal{L}(\theta_1, \theta_2, \theta_3|\{x_{i,j}\}_{i=1, j=1}^{N, M_i})=\prod_{i=1}^{N}\int_{0}^{1} f(p_i;\theta_1) \prod_{j=1}^{...
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do marginal density functions derived from a joint pdf always integrate to 1 (are they valid pdf's)?

If I have a joint pdf of multiple random variables, say 3 for simplicity, $f_{X,Y,Z}(x,y,z)$, is it true that the marginal density functions derived from that joint probability distribution ( $f_{X}(x)...
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Are there margins such that, while the "correlation" parameters of a Gaussian copula are positive, the correlations between the margins are negative?

Let there be a multivariate distribution $F$ with margins $F_1,\dots,F_n$ and a Gaussian copula with "correlation" matrix $\Sigma$. Let the off-diagonal elements of $\sigma$ be positive. Let ...
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Marginalising out two variables from an expectation

Lets say I have the following expected value: $$E[Y|X_1=x_1,X_2=x_2,X_3=x_3]$$ and I want to marginalize out the continuous random variables $X_2$ and $X_3$ to arrive at: $$E[Y|X_1=x_1]$$ How can I ...
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When transforming t in the joint probability density f (x, t), i.e. t=1/(mt+1), how does the marginal probability density f (x) transform?

for example, if: $$\begin{aligned} f_{X_{pi},T_{pi}}(x,t|*)=& \begin{aligned}\frac{\pi s^2}{\alpha_{pi}^2}\exp\left(\frac{\alpha_{pi}(x-\frac12)\nu_{pi}}{s^2}-\frac{\nu_{pi}^2}{2s^2}(t-\tau_p)\...
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Incorrect reasoning on sampling from marginals

Suppose we want to sample $X$ from a density $p(x) = e^{f(x)}$. My thought was that maybe we can introduce an auxiliary variable $Y$ such that $p(y|x) \propto e^{-\|(x,y)\|^2/2 - f(x)}$. Then the ...
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Conditional distribution $f(x|y)$ if $X$ and $Y$ are independent

Suppose we have two randome variables $X$ and $Y$, with joint distribution $f(x,y)$. $X$ and $Y$ are independent if and only if the marginal distribution of $X$ is the same as the conditional ...
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Doing empirical Bayes with improper prior - marginals that do not exist?

I am considering a Bayesian linear model for which the prior is not proper. The model is as usual $y = X \theta + w$ where $w \sim N(0, \sigma^2)$, and $\theta, \sigma^2$ are unknown. The distribution ...
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Existence of Distribution with Given Multivariate Marginals

Consider discrete random variables $X_1,\cdots, X_n$, and let $D$ be their joint distribution. For each subset $S\subseteq[n]$ let $D_S$ be the marginal distribution $(X_i)_{i\in S}$. Fix $k<n$. ...
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Marginal Gaussian distribution

I have a confusion over an integral involving a multivariate and a univariate Gaussian. We know that in the case of two multivariate Gaussians the following is true: $$ \int \mathcal{N}(\mathbf{y}|\...
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Customer default rate - joint distribution, marginal distribution when sum not equal 1

hope this question fits here, any help would be greatly appreciated. Background: I´m working on a data generator for customer default data. The default rate shall depend on multiple categorical ...
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Two dimensional random variable with uniform marginal probability density functions [duplicate]

I have access to some data for two variables - let's call them x and y. In particular, I have the distribution of data separately per each variable, something that allows me to estimate the marginal ...
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Are two marginal distributions of a student-t copula equivalent to using two independent uniform distributions?

I am trying to figure out if these two are the same: Using the marginal uniform distributions of a student-t copula Using independent uniform distributions I have generated SAS code to figure this ...
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How to obtain the following marginal distribution

I'm struggling on how to calculate the marginal $f(x)$. I'm trying to integrate $f(x,y)$ by $y$, but I couldn't solve it. The pdf is the following. $\displaystyle f(x) = \int_0^{\infty} f(x,y)\text{d}...
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Minimizing the NLL of a t-distribution derived from a NIG prior

My question concerns this paper which is a little too succinct for me to understand. The context is the following. Suppose $y$ is Normal distributed, with a Normal-Inverse-Gamma prior, $$ y \sim N(\mu,...
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Regression models with uncertainty as T-Distribution [duplicate]

Are there any distributions which represent uncertainty about $\hat y$ as a $t$ distribution. In other words: $ p(y\mid x, ~\text{model}) \sim \mathbf T(\hat y, \text{df}, \sigma)$ where $\bf T$ is a ...
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Gaussian as Markov kernels

Let the one-step transition dynamics of a system be given as \begin{equation*} \mathbf{P}[X_{t+1} | X_t] = N(f_\theta(X_t), I) \end{equation*} that is the Normal with mean a function of $X_t$ ...
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In the case of the beta-binomial model, how to compute the marginal likelihood?

I have a question about Bayesian statistics. In a book called PML, I came across the following description of how to calculate the marginal likelihood of a beta-binomial distribution. In the case of ...
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Completing the square and marginalizing a multivariate Gaussian [closed]

Edit: This question has been closed for being unrelated although I see similar questions posted here with the same objective, yet not with enough detailed answers or not exactly what I am looking for (...
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Finding regression equations from joint distribution

Let $X$ and $Y$ be two random variables with joint probability density function $f(x, y) = 1$ if $− y < x < y, 0 <y< 1$ and $0$ elsewhere. Find the regression equation of $Y$ on $X$ and ...
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Is every coordinate-wise marginal of a proper joint distribution also proper?

Suppose that we have a probability density function $\pi(x_1, \ldots, x_n)$ which is the density of a vector-valued random variable $X$ in $\mathbb{R}^n$. Here the density is proper, i.e., $\int_{\...
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Is there a simple error in the answer key, or am I using the wrong approach to get $P(X<0.5)$

I am working on a problem that gives me a joint pdf: $$f_{x,y}(x,y) = 6xy, 0<x<1, 0<y<\sqrt{x} $$ I am asked to find $P(X < 0.5)$ with three decimal places. My approach was to integrate:...
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How can I marginalize $\boldsymbol{\alpha}$ out of my hierarchical model?

Suppose I have the following hierarchical distribution: $$\mathbf{y} \sim \text{Normal}(\mathbf{X}\boldsymbol{\beta} + \mathbf{K}\boldsymbol{\alpha}, \sigma^2\boldsymbol{\Sigma}_y),$$ $$\boldsymbol{\...
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Some questions about the posterior distribution when the marginal distribution is zero

Let $\{f(\cdot|\theta): \theta \in \Theta \}$ be a family of pdfs and let $\pi: \Theta \to \mathbb{R}$ be a prior. According to Bayes' theorem (as stated in, e.g., Casella and Berger), the posterior ...
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Notation for calculating $p(v)$ marginalising over $h$ in an Restricted Boltzmann Machine

I am working through equation 22 in Introduction to Boltzmann Machines I am a little confused with the notation, in particular in the line: As I understand it, we want the probability of a specific ...
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Calculate marginal distribution using chain rule

I am trying to calculate the marginal distribution P(X) of the following joint distribution P(X, Y). y=0 y=1 y=2 x=0 .2 .1 .2 x=1 0 .2 .1 x=2 .1 0 .1 Here is how I am calculating $P(X=0) = P(X = ...
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marginal cdf and cdf of marginals

given a joint distribution of 2 variables $P(X,Y)$, is the cdf of the Y-marginal distribution equals to the Y-marginal of the cdf of $P(X,Y)$?
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Equivalent of Conjugate Priors for Marginal Probability Distributions

In probability, there are nice “conjugate prior” distributions that enable closed-form Bayesian updating – e.g. if you have a Normal likelihood and Normal prior (on the mean parameter), you get a ...
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Find marginal distributions of $f(x,y) = 1$ within shaded area A, given by boundaries $y = x^2, ~y = x^2 + 1$ and $ 0 < x < 1.$

I understand that I'm supposed to integrate with respect to x and y to find the marginal distribution of Y and X respectively, but my answers seem wrong. I got marginal distribution of $X = 1$ and of $...
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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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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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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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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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