Questions tagged [multivariate-distribution]
Probability distribution over vectors (as opposed to univariate distributions that are over numbers).
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questions
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Distribution of a product of a matrix with a random matrix
Suppose we have the matrices $Z\in \mathbb{R}^{n\times n}$ and $X\in \mathbb{R}^{n\times d}$, such that each row $x_i\in\mathbb{R}^d$ is drawn i.i.d from a $N(0,\Sigma_{d\times d})$ distribution. ...
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17 views
Your class has 100 students and they have 5 elective courses to choose from. In each course, the proportion of students is equal in population
I am actually unable to understand the question and would appreciate it if someone can help with that. For the above problem statement, there are three questions that I need to answer
Make a ...
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1answer
106 views
Variance of expectation vs. Expectation of variance
Can we compare
$Var_X[E_Y(f(X,Y))]$
and
$E_Y[Var_X((f(X,Y))]$ where $f()$ is any function of $X$ and $Y$ iid? I suspect $Var_X[E_Y(f(X,Y))]$ is the smaller one. Though I couldn't find a single counter ...
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40 views
How is this equation deduced?
These two equations are from the book Gaussian Process for Machine Learning. First we already have equation (2.8).
$p(\mathbf{w}|X, \mathbf{y}) ∼ N (\frac1{\sigma_n^2}A^{−1}X\mathbf{y}, A^{−1})$ (2.8)
...
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1answer
22 views
Is sampling from a transformed pdf the same as transforming samples in the same way from the original pdf?
Suppose I have a pdf $f(\vec{v})$ over $\vec{v}$ and define $\vec{u}=T(\vec{v})$, i.e. a transformation of $\vec{v}$. Would sampling $f(\vec{u})$ be the same as taking samples $\vec{v}_i$ from $f(\vec{...
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19 views
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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28 views
How to apply the multivariate Poisson distribution?
From my recent question here: Calculating the probability of a car accident
I wish to learn how to apply a multivariate poisson distribution in the example that car-accidents vary for each year, ...
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52 views
If normally distributed random vector $X$ has a PD covariance matrix, then any conditional distribution induced by $X$ has a PD covariance matrix?
Suppose I have a random vector $X$ whose distribution is joint normal. I know that the covariance matrix of $X$ is positive definite. I wonder if I partition $X$ in any way: e.g., $X=(X_1, X_2)$, then ...
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28 views
Mahalanobis distance?
I have a question regarding the use of the Mahalanobis distance as a measure of the goodness-of-fit of a bivariate Guassian pair.
Let's assume that we have some theoretical relationship
$$\bigg(\begin{...
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0answers
59 views
Restricting Kernel Density Estimate to n-sided polygon
Given some arbitrarily distributed data, in this case data generated by two normal distributions,
$\mathcal{N_1}(\mu_1, \Sigma_1)$ and $\mathcal{N_2}(\mu_2, \Sigma_2)$ where $\mu_1 = \begin{bmatrix}0 &...
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9 views
quantile surface of a mulitvariate distribution made of multiplication of marginal distributions assuming independence
How to perform quantile regression in a more elegant fashion?
As discussed above, quantSheets() can only deal with one explanatory variable for computing quantile ...
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0answers
19 views
Distribution of Geometric Brownian Motion drawdowns from realizations of multivariate Normal and Laplace distributions
I am trying to simulate the distribution of Geometric Brownian Motion drawdowns from realizations of multivariate Normal and Laplace distributions under the same covariance structure. Drawdowns are ...
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1answer
24 views
Sum of two different freedom degree multivariate t dsitributions
there are two multivariate t distributions whose freedom degree $\nu$ are different to each other.
$$
\mathbf{x} \sim \mathcal{T}(\nu_x, \mathbf{0}, \Sigma_x)
$$
$$
\mathbf{y} \sim \mathcal{T}(\nu_y, \...
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33 views
Testing underlying, latent normality of binary data
I have two multivariate binary datasets of 4 variables (or components) - P, Q, R and S each. Each row of the data is like a test on the components to say if they work (0) or fail (1) when tested. The ...
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2answers
131 views
How to show that $\left|\left|E\left[\frac{x-X}{||x-X||}\right]\right|\right|\xrightarrow{||x||\to\infty}1$ for a multivariate distribution?
I am trying to prove that $\left|\left|E\left[\frac{x-X}{||x-X||}\right]\right|\right|\xrightarrow{||x||\to\infty}1$ for any multivariate distribution.
My Progress: I have proved the result for ...
3
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1answer
139 views
Mean of Generalization of the Dirichlet Distribution
I know that if $X_{1},X_{2},...X_{n}$ are independent $\mathrm{Gamma}(\alpha_{i},\theta)$ - distributed variables (notice they all have the same scale parameter $\theta$) and
$Y_{i}=\frac{X_{i}}{\sum_{...
2
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1answer
60 views
Inner Product for Geometric Interpretation of Multivariate Random Vectors
I was looking into the geometric interpretation of random variables as random vectors in a vector space. The textbook I'm referring to defined $\operatorname {Cov}(X,Y)$ as the inner product for any ...
1
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1answer
35 views
PDF of a degenerate multivariable gaussian distribution
The pdf of a multivariate gaussian distribution with mean $\mu\in\mathbb{R}^n$ and variance $\Sigma\in\mathbb{S}^n_{++}$ ($\Sigma$ is positive definite) is given as
$$ p(x) = \frac{1}{\sqrt{|\Sigma|(2\...
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16 views
Conditional distribution of cross covariance
Assume M follows an Inverse Wishart distribution (known parameters), what is the conditional distribution of X given Y and Z in the below (where Y and Z are square sub-matrices)?
$$
M =
\...
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28 views
What is the sample size required to estimate a multivariate joint histogram?
The required sample size for estimating a multivariate joint histogram is something that I expect to depend on multiple factors such as the distributional properties of the data-generating process (e....
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1answer
29 views
Generate a multivariate normal vector
I'm working in R and I was wondering, let's say I want to generate a random vector $X \in \mathbb{R^p}$, with $X \sim N(0,I)$ where $I$ is the identity matriz in $\mathbb{R}^{p \times p}$. The ...
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26 views
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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0answers
11 views
How to determine if a multi-dimensional vector comes from a multi-variate distribution or not?
I have a complex multivariate distribution given by a multivariate random number generator (black box). In other words, whenever I call the "generator" I get a random vector containing ...
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36 views
Simplifying assumption for regular vine distributions
I am trying to understand vine copulas and specifically, conditional bivariate copulas. Let $c_{X_1,X_3|X_2}(\cdot,\cdot | \cdot)=:c$ be the probability density function of a bivariate conditional ...
3
votes
2answers
91 views
Can a random variable be uncorrelated with its product with a correlated random variable?
I have a random variable $X.$ I want to find a random variable $Y$ such that $Y$ is correlated with $X,$ but $Y$ is not correlated with the product of $X$ and $Y.$ Is it always possible?
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35 views
Conditional Expectation of a normal distribution [duplicate]
say we have a multivariate normal distribution with ${\boldsymbol Y} \sim \mathcal{N}(\boldsymbol\mu, \Sigma)$
The conditional expection is $\overline{\boldsymbol\mu}=\boldsymbol\mu_1+\Sigma_{12}{\...
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0answers
27 views
Test goodness of fit of a multivariate distribution via testing GOF of linear combinations of components
I am modeling a continuous bivariate distribution of a random vector $(X_1,X_2)$ using a copula. I would like to assess how well I am doing. Given a data sample, I could probably do a bivariate ...
4
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1answer
422 views
KL divergence for joint probability distributions?
I have a pair of joint probability distributions. I want to measure their similarity/dissimilarity. If they were single-dimensional probability distributions, then I could measure the Kullback–Leibler ...
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1answer
167 views
What is a generalisation of t-test to the case of multivariate distribution?
When we have a sample of numbers, one of the most basic tests is the t-test, in which we check the null hypothesis that the population mean is equal to zero.
I am interested in a generalisation of ...
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0answers
66 views
Mutual information relationship to copula entropy is borked?
I have posted a related Question based on a paper, Ma, Jian, and Zengqi Sun. "Mutual information is copula entropy." Tsinghua Science & Technology 16.1 (2011): 51-54.
In the paper, they ...
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0answers
75 views
Copula entropy: calculation is borked?
I came across a pretty cool paper whose idea makes a lot of sense to me.
Ma, Jian, and Zengqi Sun. "Mutual information is copula entropy." Tsinghua Science & Technology 16.1 (2011): 51-...
4
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1answer
433 views
What is the best way to sample points from an arbitrary 2D distribution?
I want to sample points $(x,y)$ randomly according to the Himmelblau function
$$f(x,y) = (x^2 + y - 11)^2 + (x + y^2 - 7)^2\qquad -5\le x,y\le 5$$
which I treat as a multivariate probability density ...
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0answers
19 views
Covariance matrix in multivariate is Wishart random variable
I'm working on sample variance distribution, on page 111 of Methods of Multivariate Analysis, it says
"The joint distribution of these $p(p + 1)/2$ distinct variables in $W =(n−1)S = \sum_i(y_i − ...
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33 views
Is it reasonable to look at the output of simulating from a multivariate distribution as univariate distribution? If yes, what is this called?
Suppose I have $X_{n} \sim MVN(\underline{\mu},\Sigma)$ where $n$ is large (several thousands). However, the $\mu_i's$ and the elements of $\Sigma$ are such that almost every simulation from $X_n$ ...
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0answers
29 views
Measuring entropy/concentration in multivariate datasets
I am interested in measuring how concentrated or widely dispersed a certain binary property is among a population, which is defined by a number of categorical variables.
For instance, let's say I have ...
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0answers
49 views
Multivariate Bernoulli Logistic Regression on OpenBUGS
I am new to WinBUGS/OpenBUGS, and am trying to solve this Multivariate Bernoulli Logistic Regression
$$P(θ1,θ2|X,Z) ∝ P(X|θ1) P(θ1) P(Z|X,θ2) P(θ2)$$
Is this valid on OpenBUGS?
This is my OpenBUGS ...
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0answers
14 views
Does the t-copula capture serial nonlinear dependence?
Say $\mathbf{Y}=[y_1,\cdots,y_n]'$ with elements that are uncorrelated, yet serially dependent. As a result, in the literature, the means of allowing for nonlinear serial dependence for processes ...
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0answers
26 views
PDF for Bayesian Decision
I have the discrete-time sequence $X(n) = 1$ with probability 3/4 and $X(n) = 0$ with probability 1/4 for $n = 1,...,10$. The observed signal is $Y(n) = X(n) + N(n)$ where $N(n)$ is an AWGN with ...
4
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2answers
139 views
Interpretation of multivariate conditional gaussian function form?
I've been reading over this Multivariate Gaussian conditional proof, trying to make sense of how the mean and variance of a gaussian conditional was derived. I've come to accept that unless I allocate ...
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0answers
46 views
Kth Order statistic of a multivariate distribution
The Kth order statistic for a univariate is equal to its kth-smallest value. For instance, given $\{6,9,3,8\}$, the 2nd-smallest value would be the 2nd order statistic.
How does this concept ...
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0answers
53 views
Multivariate Gamma parameter estimation
Consider $X$ a d-dimensional random variable with positive values, mean $\mu\in\mathbb{R}_+^d$, and covariance matrix $\Sigma\in\mathbb{R}^{d\times d}$. If I have $n$ samples $\{ x_1, ..., x_n \}$ ...
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45 views
Derive expectation of the log determinant of a precision matrix from a Wishart distribution
I'm reading through section 21.6 of Murphy's Machine Learning: A probabilistic perspective where they derive the variational bayes algorithm for fitting a mixture of gaussians. One of the steps ...
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16 views
plotting bivariate normal distribution samples
I have generated two samples $\underline{X_i}$ and $\underline{Z_i}$
$\underline{X_1}$ $\underline{X_2}\dots \underline{X_{5000}}$ , while $\underline{X_i} \sim N_2[(1,2)^T,\begin{pmatrix}2&1.5\\ ...
2
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2answers
58 views
Why is there covariance instead of simply variance in multidimensional normal distributions?
Maybe it is just because I'm only experimenting with 2 dimensional normal distributions, but multi-dimensional normal distributions for me seem like just multiple one dimensional normal distributions.
...
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0answers
59 views
Probability distribution associated with nuclear norm?
The $\ell_1$ norm of model parameters is often added to loss functions because it induces sparsity in the solution of the overall cost function:
$$ c(\theta) = \log L(x|\theta) + \lambda ||\theta||_1$$...
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0answers
14 views
Condition Random Variable on Range of Another Random Variable [duplicate]
Assume that $v_s \sim N(\mu_s,\sigma_s^2)$ and $v_b \sim N(\mu_b,\sigma_b^2)$, denote their correlation by $\rho$, and assume they are jointly normally distributed. How would I assess $E[v_b|v_s\leq c]...
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16 views
Bivariate random effect problems in selection models (Mixture Cure model)
I am currently working on a mixed effects selection model.
The selection model is a logistic model with a Gaussian random effect.
The principal model is a survival model with a Gaussian random effect (...
2
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1answer
168 views
Find expectation of conditional normal distribution
I am struggling with some finding expectation value question .
the question is to find $E[Y|X]$ from the result $P(Y|X)$
with given mean and covariance
$$\mu=[\mu_x, \mu_y]^T$$
$$\Sigma=\begin{bmatrix}...
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0answers
31 views
What is the entropy of multivariate data multiplied by a vector?
It is a general rule that for multivariate data $\boldsymbol{X}$ and a matrix $\boldsymbol{A}$, their entropy is
$$h(\boldsymbol{A} \boldsymbol{X}) = h(\boldsymbol{X}) + \ln |\det \boldsymbol{A}|$$
(...
2
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0answers
194 views
What is the distribution of the CDF of a sample drawn from a multivariate normal?
Introduction: Lets say we have a random variable $X$ that follows a normal distribution, $X \sim N(\mu, \sigma^2)$ , with a CDF function $F_X(x) = P(X \leq x)$. Then we draw some random samples $S_1$, ...
