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 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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Median marginalized likelihood: how to compute that?

I want to compute the median marginalized likelihood of a parameter. I have a grid in which: rows refer to models, and columns refer to each parameter. I should estimate the median ML of one of these ...
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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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Computing factor function in a factor graph

Consider a factor graph with three binary variables X1, X2, X3 connected to one hard constraint factor f whose compatibility function imposes a OR function, i.e. f(x1, x2, x3) = 0 if x1 = x2 = x3 = 0, ...
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How can I practically “marginalize away a nuisance parameter”?

I generate my data with this model: $y=ax+b+\nu$ where $\nu$ is a random value from a random variable which follows a normal distribution with mean equal to zero and standard deviation equal to $\...
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Testing for marginal independence

I have a dataset which consists of ~200 lists each with 2-8 measurements. For example, four of the lists might look like this: $L_1 = \{0.31, -0.56, 3.12, -0.25, 0.67\}\\ L_2 = \{0.11, -0.43\}\\ L_3 = ...
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Deriving the marginal multivariate Dirichlet distribution

I am trying to understand how my professor (see derivation below) has derived the multivariate marginal distribution of a subvector of $\theta_j$´s from a Dirichlet distribution. I understand ...
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Difference between multiplied marginal probabilities and joint probability?

For real-valued data $X$ and $Y$, what is the difference between $$p(x,y) \quad \text{and} \quad p(x) p(y)$$ where their joint probabilities are (left), and multiplied probabilities (right), otherwise ...
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Bias in unbiased pseudo-marginal estimation

In the Pseudo-marginal Metropolis-Hastings algorithm exact sampling of a posterior distribution is performed when using an unbiased estimate of the marginal likelihood. However, I am having problems ...
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Horizontal vs Vertical optimal transport of probability distributions

Optimal transport has a primal and dual form. My question is: Is the primal formulation of OT intended for horizontal movement of probability mass, whereas the dual formulation is more geared towards ...
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Marginal Distribution for x [closed]

I want to calculate the marginal distribution of $X$ given that the joint probability density function of $(X,Y)$ is given by $$f(x,y)=2592(x^2-y^2)e^{-2x} \qquad 0<x<\infty,\ -x<y<x$$ My ...
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Find the MARGINAL survival function

Find the marginal survival of $X$ when $$ S(x,y) = (1-x)(1-y)(1+\frac{xy}{2}),0<x<1,0<y<1$$ So if we have a joint pdf $f(x,y)$, then the marginal is $f(x) = \int_{-\infty}^\infty f(x,y)dy....
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Hypothesis test for discrete random vector with few samples

Consider a random vector $X \in \mathbb{R}^{d}$ with support $\text{supp}(X) = \{1,2,3,4\}^d$, and let $P_X$ denote its known probability mass function. Note that $\lvert \text{supp}(X) \rvert = 4^d$. ...
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Behaviour of the marginal in the limit for an infinite sequence of hierarchical priors

Consider the following model: $$y \sim \text{Exponential}(\lambda_0) \\ \lambda_i | \lambda_{i+1} \sim \text{Exponential}(\lambda_i+1) \\ \text{for } i=1,2,\dots,d\\ \lambda_{d+1} = k $$ With an ...
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the law of total probability with extra variables

Suppose $X$ and $Y$ are two discrete random variables. The law of total probability states that: $$ p(x) = \sum\limits_y {p(x,y) = } \sum\limits_y {p(x|y)p(y)} $$ Now suppose we have another random ...
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What does the normalization factor mean in Bayesian learning?

Suppose we want to learn the distribution of a random variable $x$. Assuming it's from a Gaussian distribution: $x\sim\mathcal{N}(\theta,1)$ with constant variance 1, we can learn the parameter $\...
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If sampling from parametric copula fitted to real data, must the marginals also be simulated?

A real dataset consists of two random variables $X$ and $Y$ whose densities are the marginal distributions to be used. These marginals are transformed into uniform variables. As for the copula, either ...
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Integrating the product of two Gaussians with different dimensions

I'm relatively unfamiliar with sort of integration calculus, so apologies in advance for any notation issues. Given parameters of some linear-regression model $\mathbf{a}=(a_1,a_2,a_3)$, and ...
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Solving the marginal probability given the probability distributions

Given the probability distributions as follows $P(b_1|a_0,c_0)=p$ $P(b_1|a_1,c_0)=o$ $P(b_1|a_0,c_1)=n$ $P(b_1|a_1,c_1)=m$ $P(a_1|c_1)=x$ $P(a_1|c_0)=y$ $P(c_1)=r$ I need to find $\dfrac{P(b_1|a_0)}{P(...
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Kullback-Leibler divergence between marginals and joint distribution? (It's not mutual information)

Mutual information is defined as the Kullback-Leibler divergence between a joint distribution and its marginals: $$I(X,Y) = \mathrm{KL}(P(x,y)||P(x)P(y)) = \sum_{x,y}P(x,y)\ln\left(\frac{P(x,y)}{P(x)P(...
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Some questions regarding Conditional Probability and hypothesis testing?

I know that P(B|A) = P(A and B) / P(A) by Bayes Rule, but what happens if A can vary across two values such as A = 1 and A = 0? How then would I find P(B|A)? Eg. If I had a scenario where P(B) = the ...
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Is probit transformation the same as probability integral transform?

The image shows the original marginal data $u$ and $v$ on the left, which has a bounded support, and their probit transformations $r$ and $s$ on the right, which has an unbounded support. The ...
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Poisson distribution marginal probability of sufficient statistic

I am self studying a theoretical statistics course I found online. There is a question to show that for $(X_1, ... X_n)$ i.i.d Poisson variables with parameter $\theta$, the statistic $T=\sum_{i=1}^N ...
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How to measure the level of tail dependence in copula?

In the Clayton copula below, we see that there is stronger lower tail dependence (bottom left corner of Clayton) than upper tail dependence (upper right corner) because the pseudo-observation pairs in ...
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Marginalizing multivariate Gaussian distribution

While working through the exercises in Mathematics for machine learning I have encountered a claim (Eq. (6.68)) that the marginal of a two-dimensional normal distribution $\mathcal{N}(x, y |\mathbf{\...
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35 views

How can I get probability for an interval with continuous marginal probability function?

Let's say, I have joint probability function as follows: $ f(x,y) = 4xy $ for $ 0 \le x \le 1 $ and $ 0 \le y \le 1 $ I want to get the marginal probability distribution of the random variable $ X $ ...
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Derivation of Sklar's theorem for copula

The joint probability distribution, or bivariate CDF, of two random variables $X$ and $Y$ is $$F(x,y) = P(X\leq x, Y\leq y)$$ where $F(x) = P(X\leq x)$ would be the marginal distribution of $X$. Sklar'...
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What are the differences between “Marginal Probability Distribution” and “Conditional Probability Distribution”?

While studying probability, I am kind of having difficulties in understanding marginal probability distribution and conditional probability distribution. To me, they look much the same and cannot find ...
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What is the support of a copula, vs support of a marginal?

Continuous marginal distributions have a potentially unbounded support. (does unbounded support mean $[0,\infty)$ as much as it could mean $(-\infty,\infty)$?) Is it true then that, on the other hand, ...
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What is the copula of a variable with itself?

In Sklar's theorem for joint probability functions, $$f(x,y) = c(F_X(x), F_Y(y)) \cdot f(x) f(y)$$ the copula is $c(\cdot)$ of variables $X$ and $Y$, while $f(\cdot)$ are their marginal distributions. ...
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How to describe an “incomplete” prior?

I would like to know how to describe sources of uncertainty neglected when I approximate a prior distribution $p(x)$ by a marginal distribution. Specifically, let's say that I have a marginal ...
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147 views

Kullback-Leibler divergence and marginals

Let $P(x,y)$ and $Q(x,y)$ be two probability distributions, with marginals $$P(x)=\sum_y P(x,y),\quad P(y)=\sum_x P(x,y),\quad Q(x)=\sum_y Q(x,y),\quad Q(y)=\sum_x Q(x,y)$$ What is the relation ...
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Gaussian Mixture Model $p(x_i | z_i = k)$ a likelihood or probability?

In Gaussian Mixture models, the probability of observing the data $x$ given that it was generated from $M$ gaussian models is given by the following equation $$p(x) = \sum_{k=1}^m p(x|z=k)p(z = k)$$ ...
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In these domains do they have different conditional probability distribution AND marginal probability distribution?

For simplicity, I'm going to focus on subject 1 and subject 4 and only observe class 3 (green) and class 2 (blue), here's my understanding: The have different conditional probability distribution, ...
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Existence of marginal of time-varying joint distribution

In this paper, in Definition 3 page 4, the authors define the temporal coupling $\gamma_{s,t} := \mathcal{L}(X_s, X_t)$ as the law of the joint distribution of $\mathbb{P}_s$ and $\mathbb{P}_t$. They ...
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56 views

How to identify a vine copula?

In the image below is a mixed vine copula composed of 3 copulae, therefore 3 marginals. It shows the 2d marginals and 100 samples of a 3d mixed vine. Without being told its a vine copula, how would ...
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If the joint density $f_{X_1,…,X_n}(x_1,…,x_n)$ is symmetric about the origin, does this imply that each marginal cdf $F_{X_i}(0)=1/2$?

If the joint density $f_{X_1,...,X_n}(x_1,...,x_n)$ is symmetric about the origin in the sense that for any $(x_1,...,x_n)$, it holds that $f_{X_1,...,X_n}(x_1,...,x_n)=f_{X_1,...,X_n}(-x_1,...,-x_n)$ ...
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Frequentist approach to marginalize nuisance parameters

How would be a frequentistic approach to solve this problem? "We have a random machine that gives 0 or 1 with a hard-coded, fixed but unknown probability $p$. After 10 trials we have 5 "0&...
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Estimating conditional probability P(A,B)/P(B) by using separate models for P(A, B) and P(B) vs marginalisation of P(A,B)?

edit: Tried to clarify the question. Say I want to estimate $\frac{P(B | A) P(A)}{P(B)}$, and for some reason I can't estimate $P(A|B) = \frac{P(B | A) P(A)}{P(B)}$ directly. For the numerator, I can ...

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