Questions tagged [multivariate-distribution]
Probability distribution over vectors (as opposed to univariate distributions that are over numbers).
155
questions
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16 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 ...
0
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0answers
6 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
7 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
25 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 ...
5
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2answers
107 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 ...
1
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0answers
21 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
15 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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13 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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0answers
13 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
25 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.
...
4
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0answers
37 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
10 views
Help in calculating diagonal covariance matrix for generative model for binary classification
I am given this data. I want to fit a generative model $\cal{N}(\mu_0, \sigma_0^2 I_2)$, $\cal{N}(\mu_1, \sigma_1^2 I_2)$ for the $0$ and $1$ classes respectively using $\textbf{MLE}$ and plot ...
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0answers
10 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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0answers
53 views
Multivariate Normal distribution Cramer-Rao bound
How to compute the Cramer-Rao lower bound for $\mu$ of a multivariate normal distribution $\mathcal{N}_{p}(2\mu,\Sigma_0)$.
Also I would like to know the minimum variance unbiased estimators for $\mu$....
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0answers
10 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
votes
1answer
79 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
$$\mu=[\mu_x, \mu_y]^T$$
$$\Sigma=\begin{bmatrix}...
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0answers
24 views
What is the entropy of multivariate data multiplied by a vector?
It is a general rule that for multivariate data $\boldsymbol{X}$ and a matrix $\boldsymbol{A}$, their entropy is
$$h(\boldsymbol{A} \boldsymbol{X}) = h(\boldsymbol{X}) + \ln |\det \boldsymbol{A}|$$
(...
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0answers
48 views
What is the distribution of the CDF of a sample drawn from a multivariate normal?
Introduction: Lets say we have a random variable $X$ that follows a normal distribution, $X \sim N(\mu, \sigma^2)$ , with a CDF function $F_X(x) = P(X \leq x)$. Then we draw some random samples $S_1$, ...
2
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1answer
51 views
What is the fisher information matrix of the multivariate t distribution?
$\newcommand{\bx}{\mathbf{x}}$
$\newcommand{\bSigma}{\boldsymbol{\Sigma}}$
$\newcommand{\bE}{\mathbf{E}}$
$\newcommand{\bD}{\mathbf{D}}$
Consider the multivariate central t distribution with p.d.f.
\...
14
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4answers
2k views
Is this causation?
Consider the following joint distribution for the random variables $A$ and $B$:
$$
\begin{array}
{|r|r|}\hline & B=1 & B=2 \\
\hline A=1 & 49\% & 1\% \\ \hline A=2 & 49\% & 1\...
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0answers
7 views
Is the mvt (multivariate t) adjustment for multiplicity appropriate for repeated measures and longitudinal trials?
As in the title. If I have a longitudinal experiment with t0...tn timepoints, and want to verify the Dunnett contrast (all versus baseline), can I use the mvt adjustment for multiplicity? I think it ...
3
votes
1answer
79 views
Correlation Coefficient for Hypergeometric-type Distribution
Problem Statement: A box contains $N_1$ white balls, $N_2$ black balls, and $N_3$ red balls
$(N_1+N_2+N_3=N).$ A random sample of $n$ balls is selected from the box (without replacement).
Let $Y_1,Y_2,...
2
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1answer
192 views
Probability distribution of the product of two dependent random variables
It is well known that being $X$ and $Y$ two independent random variables with distributions $f_X(x)$ and $f_Y(y)$, respectively, then the probability distribution of the multiplicative function $z = ...
3
votes
2answers
104 views
What does i.i.d. mean for multivariate case?
When we say a random variable is i.i.d., it's often used to describe the dependency between the observations of that random variable, which I call the row dimension, indexed by time if it's a time ...
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0answers
20 views
How does the maximum entropy principle affect joint entropy, mutual information, and other info measures?
The maximum entropy principle says that we should use the probability distribution of a univariate dataset that has the highest level of entropy because it offers the lowest information. How does the ...
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0answers
54 views
Rescaling multivariable normal pdf and normalizing constant
I am trying to understand change of variables for a random variable and how it changes the pdf and the normalizing constant.
Let $\mathbf{x}$ be $N$-dimensional normal variable and let $\mathbf{\...
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0answers
26 views
What are the best known techniques to verify that a GAN samples correctly from a given distribution?
I would like to know what are the best known techniques to check that a generative adversarial network (GAN) samples from the correct distribution.
Naively, I would say it all boils down to a ...
0
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0answers
18 views
Multivariate rare events
My data is something like this:
I have U urns, and I have taken a bag of $n$ objects from each urn. Each urn has $N$ objects, and I have sampled $n$ with replacement.
$n$ is comprised of coloured ...
2
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0answers
36 views
Algorithm to generate non normal (discrete and continuous) correlated variables using a correlation matrix
What I am trying to do here is, given a dataset (let's say n observations of N variables), and thus a correlation matrix M that result from said dataset, I would like to write a Python algorithm that ...
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0answers
32 views
Empirical conditional density of continuous variables
I have a dataframe, with data of several continuous variables. The variables are not independent.
My goal is to sample from the distribution that generated this data.
What's a relatively easy and ...
0
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1answer
53 views
Multivariate distributions
Let $x$ and $y$ be random variables with the following joint density function:
$f(x,y) = e^{-x}$ for $0<x< \infty$, and $0<y<1$
If $z= x+2y$, what is the joint density function of $x$ and ...
1
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1answer
12 views
How to choose what feature vector to plot in multivariate regression analysis?
I'm new to the field of machine learning and I have been having this doubt for a long time now. If we want to plot a scatter plot, we plot it as, x as a function of y where x is a 1-D array. But in ...
1
vote
1answer
413 views
Does the covariance of i.i.d. random vectors/multivariate random variables have any zero terms?
If we have i.i.d. random variables, $X$ and $Y$, then $\text{Cov}(X,Y)=0$.
But let's say we have i.i.d. random vectors $\boldsymbol{X}$ and $\boldsymbol{Y}$, where $\boldsymbol{X}=(X_{1},...,X_{p})$ ...
1
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0answers
43 views
How to draw huge high-dimensional normally distributed vectors in a memory-efficient way
I would like to simulate a few hundred multivariate normally distributed vectors $H$ of dimension $10^6$ with covariance matrix $\Sigma$, preferrably in R. The individual entries of $\Sigma$ are ...
0
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1answer
120 views
What's the best function to test multivariate normality when sample size more than 3000?
Before MANOVA, I need to test multivariate normality.
Then I tried MVN::mvn() function in R, output as below:
...
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0answers
14 views
bivariate conditional joint sensor model
I am struggling to find $P\left( V_t | z \right)$ from $P\left( V_t | z , V_p \right)$. Here $z$ and $V_p$ are independent variables.
...
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0answers
34 views
Calculate fifth and sixth polynomials for Headrick (2002) method for non-normal multivariate distribution
I am trying to perform a 3-variable correlated multivariate Monte Carlo simulation. As the asset class returns are non-normal, I found the following function ...
0
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0answers
22 views
Likelihood of three dimensional data
I'm having a lot of trouble finding the likelihood and log-likelihood of $\tau$ that corresponds to the following equation:
$a_i = y(t_i, \tau) + \epsilon_i$
where $\epsilon_i \sim \mathcal{N}(0, \...
1
vote
1answer
25 views
Normal random matrix as linear transformation of standard normal random matrix?
Wikipedia has the following definition of a normal random vector:
A real random vector $\mathbf{X} = (X_1,\ldots,X_k)^{\mathrm T}$ is called a normal random vector if there exists a random $\ell$-...
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0answers
31 views
How to learn dependency of variables from data?
I have a data set $X$ that consist of $m$ vectors $\vec{x}$ of $n$ real valued components. Each vector component lies within a corresponding predefined interval of valid values, which is the same for ...
0
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0answers
93 views
Probability density of conditional multivariate distribution [duplicate]
We have a multivariate normal vector ${\boldsymbol Y} \sim \mathcal{N}(\boldsymbol\mu, \Sigma)$. Consider partitioning ${\boldsymbol Y}$ into
$${\boldsymbol Y}=\begin{bmatrix}{\boldsymbol y}_1 \\
{...
2
votes
2answers
45 views
How to prove a multivariate r.v. does not follow the nonparanormal distribution?
Background
You may find the definition of the non-paranormal distribution at the 2nd paragraph in p.2296 of this paper.
In short, $(X_1, \ldots, X_p)$ is non-paranormal if there exists a set of ...
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0answers
11 views
Modelling probability distribution of multivariate dataset
my background is not in stats so apologies if this is an obvious question.
I have a dataset with 60,000 observations, and 128 features. Each feature follows a hurdle distribution with either a gamma, ...
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0answers
41 views
Distribution of quadratic forms in mixed model
I have a question related to the distribution or asymptotic distribution of quadratic forms that arise in the linear mixed model. Suppose,
$$Y=X\beta + H\delta + \epsilon$$ where $\epsilon \sim N_n(0,...
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1answer
48 views
Conditional Distribution Multivariate Normal Distribution [closed]
Let $X_1, X_2, X_3$, be jointly distributed according to a multivariate normal distribution.
$[X_1, X_2, X_3]^T\sim N(\mu=[0,0,0]^T , \Sigma = [[5,0,0],[0,2,1], 0,1,3]])$
$U = X_1 + 2X_2$ and $V = ...
0
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0answers
12 views
Calculating the credible interval of each variable in a multivariate distribution
I have a k-dimensional Dirichlet posterior with parameters $\alpha_1,...,\alpha_k$ and variables $\theta_1,...,\theta_k$. This posterior comes from a Dirichlet prior and a multinomial likelihood.
...
4
votes
1answer
265 views
Multivariate normal distribution transformation
Suppose that $X $ has a multivariate normal distribution
$X\sim MVN (\mu, \Sigma) $,
How can I transform $X$ into $Z$ so that $Z\sim MVN(\mu, I) $ where $I$ is the identity matrix?
For instance, ...
0
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1answer
23 views
DeGroot P.155 integration problem for multivariate distributions
I am stuck with the integral for equation 3.7.4 and do not see how it was done. Could someone provide me with some hints or resources to read around?
0
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1answer
102 views
Confusion over multinomial and multivariate- hypergeometric distributions
You, your parents, your sister, go to visit grandma for her birthday. Grandma made a cake for the party. If she puts $20$ raisins in the cake at random in the cake, and she divides the cake into $5$ ...
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0answers
45 views
Simulations using correlated random numbers from Multivariate Normal and fat-tailed distributions
This question makes use of the LaplacesDemon package in R, but it is not a coding question, so I believe this is the most appropriate forum.
First, the unsurprising results. I generate k correlated ...
