Questions tagged [multivariate-normal]

The multivariate normal distribution, is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. (Also called, multivariate Gaussian)

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How to perform joint inference on multivariate normal variables?

Suppose I have the following model: $$\begin{aligned} \text C &\sim \mathcal N \left(\mu, \delta^2\right) \\ \forall i: \text L_i | \text C = c &\sim \mathcal N \left(c, \lambda_i^2 \right) \\...
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How is the mahalanobis distance like the euclidean distance? [duplicate]

Let's say $\vec{x}$ is an $n$ dimensional observation, $\vec{\mu}$ the $n$ dimensional mean of the sample that $\vec{x}$ is from and $\Sigma$ the $n \times n$ covariance matrix of that sample. Then ...
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Suggest a model for this dataset

I have a time series data set (the Old Faithful geyser data available here: http://www.gatsby.ucl.ac.uk/teaching/courses/ml1-2012/geyser.txt). Plotting the eruption duration on the x axis and the ...
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Sum of Normal Random Variables | Different Dimensions

I would like to generate one random numer $single\sim N(0,1)$ and create the vector that contains only this one number: $one = [single, single, ..., single]$ . Later, I would like to combine with ...
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Multivariate bayesian inference: learnig about the mean of a variable by observing another variable

I want to derive a Bayesian learning procedure where I don't only learn from my own signal, but also from other signals which are correlated to mine. I thought it could simply work with Bayesian ...
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Shape of parameters marginal posterior in hierarchical Bayes model

Consider a generic hierarchical Bayes model with data $y_i\sim p(y|\theta_i)$, dependent of parameters $\theta_i\sim p(\theta|\phi)$ and hyperparameters $\phi\sim p(\phi)$. Furthermore, assume that $\...
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Degenerate multivariate normal in Maximum Likelihood Estimator (Akaike's Info Criterion, BIC, LR Test usage)

Let's suppose that the considered set of random variable has a covariance matrix which is psd. Therefore the Gaussian pdf must be written in its degenerate form, where the determninat of the ...
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Fitting a Gaussian to two Gaussians [duplicate]

Let's say I have a dataset set of 2D points. I break the dataset to 2 subsets, equal in number. Then I fit a bivariate (2D) Gaussian to each subset. So I have two bivariate gaussians, each with its ...
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Problem with two correlated random normals

Imagine you have a two-dimensional multivariate normal random variable with $\mu = [0, 0]$ and $\Sigma\ = \begin{bmatrix}1 & r\\r & 1\end{bmatrix}$. (Conceptually, you have two random normal ...
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38 views

Delta method for vector valued functions

Suppose I have an estimator $B\in\mathbb{R}^m$ converging to $\beta$, such that $$ \sqrt{n}(B-\beta)\rightarrow\mathcal{N}(0,\Sigma). $$ I am interested in a quantity $\mathbf{h}(B):\ \mathbb{R}^m\...
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Conditional distribution of a function of a random vector given conditional distribution of random vector

Let $\mathbf{X}=(X_1,...,X_n)^T$ be a multivariate normal distribution. Now we have $\mathbf{Y}=(Y_1,...,Y_n)^T$ defined by $Y_i = e^{X_i}$. Let $\mathbf{Y^1}, \mathbf{Y^2}$ be partitions of $\mathbf{...
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Computing probability density function at a point, given the covariance matrix and mean

(Edited for clarity.) Say I have the variance-covariance matrix $\mathbf{V}$ and mean $\mathbf{\mu}$ of a multivariate normal distribution. Given a sample, $\mathbf{s}$, can I compute/estimate the ...
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Conjugate prior for DPGMMs using Gibbs sampling

I am using Gibbs sampling to infer DPGMMs. The prior for multivariate Gaussians is Normal-inverse Wishart. But it turns out that the covariances are not estimated accurately. Here is codes and results....
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Conditional probability density function (PDF) of bivariate normal distribution

Let $X$ and $Y$ have bivariate normal PDF with correlation coefficient $\rho$, i.e.,: $f(x,y)=\frac{1}{2\pi\sigma_X\sigma_Y\sqrt{1-\rho^2}}\exp{(-\frac{1}{2(1-\rho^2)}[\frac{(x-\mu_X)^2}{\sigma_X^2}+\...
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Explanatory variable independent of the response, yet has non-zero beta

My intuition was that if an explanatory variable is independent of the response then in a multiple regression it should have a $\beta$ of zero. Consider however the following very simple example: the ...
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How to characterize the effect of $(\textrm{Diag}(\Sigma^{-1}))^{-1}$ badly approximating $\textrm{Diag}(\Sigma)$

I have an almost singular covariance matrix $\Sigma\in\mathbb{R}^{n\times n}$ that has a few large eigenvalues, followed by many many comparatively very small ev's. If I were to try to approximate ...
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Compute the mean and std of a random variable from its correlation with known random variables?

Context I'm running statistical simulations on IQ distribution among people and I would like to sample the IQ of a child given : The general distribution of IQ following : $$ \mathcal{N}(100, 15.5) ...
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Multivariate normal error with autocorrelation in second dimension

I am trying to forecast my model, but am unsure how to so in terms of error distribution (using mvrnorm). The model itself essentially estimates numbers over time and state (a matrix time x state). ...
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Deriving the conditional distribution of a multivariate normal, for inequalities

This question is slightly related to Deriving the conditional distributions of a multivariate normal distribution. In that question, the following situation was given. If $Y$ follows a multivariate ...
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27 views

Concentration inequality for max component of a multivariate Gaussian in the general case

I am looking to bound the variance of the maximum component of a vector distributed multivariate Gaussian in the general case where the Gaussian distribution has arbitrary mean and full covariance ...
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Integrating with a multivariate Gaussian

I need to figure out the steps to solve the following integral, where $Q(\mathbf{w})$ is a multivariate Gaussian with mean $\overline{\mathbf{w}}$ and covariance $\mathbf{C}$: \begin{align}\int Q(\...
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1answer
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Estimate parameters of a normal knowing the density function

I have a matrix with 3 columns. The first column contains values of a variable $x_1 \in [-1,1]$. The second column contains values of a variable $x_2 \in [-1,1]$. The third column contains a variable $...
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Distribution of $XY$ when $(X,Y) \sim BVN(0,0,1,1,\rho)$

The question is pretty much in the title, I need to find an approximate distribution of $XY$ when $(X,Y)$ follow a Bivariate Normal Distribution where $X$ and $Y$ are each $N(0,1)$ distributed and $...
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Simplification of bivariate normal $\phi_2(x,y,\rho)$ at $y=y_F$ (i.e. fixing one of the axes)

Suppose we start off with the traditional standard bivariate normal distribution: $$\phi_2(x,y|\rho,\mu_x=0,\mu_y=0,\sigma_x=1,\sigma_y=1)=\frac{1}{2\pi\sqrt{1-\rho^2}}\exp \left(-\frac{x^2-2\rho x y ...
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Using Hotelling's T-statistic to find an elliptic confidence set

The problem: We have samples of sizes ${n_1} = 25,{n_2} = 15,{n_3} = 30$ drawn independently from $N\left( {{\mu _i},{\sigma ^2}} \right),i = 1,2,3$ (normal distributions with same variance). We have $...
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Phase marginal for a multivariate complex Gaussian density

Suppose $z$ is a random variable taking values in $\mathbb{C}^n$ and admitting the complex Gaussian density $p(z;W) \propto \exp{(-\frac{1}{2}z^*Wz)}$, where $W$ is Hermitian. Let $r$ be the vector of ...
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Multivariate Theory: How does the new mean only depend on the conditioned variable?

I'm doing some review of Gaussian Processes and Multivariate Normal Theory. I found a really helpful website here, but I have run into a snag. What does the author mean in the sentence below this ...
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32 views

Conditional covariance of multivariate normal tail

Let $X\sim N(\mu,\Sigma)$, $t\in\mathbb{R}$, and $a$ be a non-zero vector of the same dimension as $X$. Define a random vector $Y=X\mathbb{1}(a^\top X\ge t)$, where $\mathbb{1}$ denotes the indicator ...
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Computing posterior based on sum of multivariate normal distribution

Currently I am exploring topics for my undergrad thesis. Although I took a course in Bayesian statistics, I am not yet sure how to proceed in finding the posterior in the following case. I have a d-...
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66 views

Distribution of transformed multivariate log-normal

Let $\mathbf{X} \sim \mathcal{N}(\boldsymbol{\mu}, \Sigma)$ and $\mathbf{Y} = \text{exp}(\mathbf{X})$. If $Y_i$ is one of the components of $\mathbf{Y}$, what is the distribution of $\frac{\mathbf{Y}}{...
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Validity of confidence interval for $\rho$ when $X\sim N_3(0,\Sigma)$ with $\Sigma_{ij}=(\rho^{|i-j|})$

Suppose $X\sim N_3(0,\Sigma)$, where $\Sigma=\begin{pmatrix}1&\rho&\rho^2\\\rho&1&\rho\\\rho^2&\rho&1\end{pmatrix}$. On the basis of one observation $x=(x_1,x_2,x_3)'$, I ...
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Taking constant out of multivariate normal

For a univariate normal distribution $X \sim N(0, k\sigma^2)$ we can take out the $k$ to get $\sqrt{k}X \sim N(0, \sigma^2)$. In the multivariate normal case is there something similar? If $\textbf{Y} ...
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Asymptotic covariance matrix of the maximum likelihood estimator of the parameters of a multivariate normal distribution

I would like to know how to derive the asymptotic covariance matrix of the maximum likelihood estimator of the two parameters (mean vector and covariance matrix) of a multivariate normal distribution. ...
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1answer
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proof of independence of X-Y and X+Y when X,Y come from bivariate normal

I have a bivariate normal distribution: $$\begin{pmatrix} X\\Y\end{pmatrix} \sim \mathcal{N}(\mu, \Sigma)$$ where: $$\mu = \begin{pmatrix}\mu_X \\ \mu_Y\end{pmatrix}$$ $$\Sigma = \begin{bmatrix} \...
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1answer
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How to evaluate multivariate normal integral with conditional upper bounds

Suppose I have independent normally distributed random variables: $x_i \sim N(0,1)$. In my actual application, $i=1,\ldots,30$, but for my example here I'll use $i=1,2,3$. I want to evaluate (either ...
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R2jags: Truncated Multivariable Normal Distribution in JAGS model (three variables)

I’m trying to specify the joint distribution of 3 parameters in JAGS using a truncated multivariate normal (one of the 3 parameters is truncated at zero, the others are untruncated). I’ve ...
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Derive multivariate from bivariate normal distribution

Could anyone help me on the following. Let $K$ and $M$ be integers so that $K\geq3$ and $2\leq M < K$. Let $\boldsymbol{X}=(X_1, ..., X_K)^T$ be a random vector, $\boldsymbol{\mu}$ be a $K\times 1$...
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1answer
43 views

Fitting a multivariate Gaussian with extremely sparse samples

We have a multi-variate Gaussian distribution. For instance with 3 variables. The correlations between the variables are important! We are fitting it to data, however, the samples are such that each ...
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1answer
86 views

Observed information matrix with multivariate normal distribution

$$ \DeclareMathOperator\tr{tr} \DeclareMathOperator\vecOP{vec} \newcommand\di{\mathrm{d}} \newcommand\D{\mathrm{D}} \newcommand\Hess{\mathrm{H}} $$ I do not have much experience with matrix ...
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Help with normalising data that has A LOT of 0s [duplicate]

I recently am analysing my results (behavioural, observation-based data), and I realised that my data are non-normal. No problem, this happens in behavioural data a lot, and I thought I just needed to ...
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1answer
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Does standard distance follow the 68-95-99.7 rule?

I'm wanting to do a simple standard distance demonstration for my students in R, but I've come across a conundrum. When I simulate the creation of 10,000 points in a spatial normal distribution, ...
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1answer
54 views

Finding the marginal distribution

If $p(y) =N(0,\Gamma) $ and if $p(x|y) =N(\Lambda y, \psi I) $where $I$ denotes the identity matrix. Also, we have the conditions: $\Gamma_{ij} =0$ for $i \neq j$ (diagonal) $\Lambda \Lambda^T = I$ ...
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Expectation of multivariate gaussian w.r.t. other multivariate gaussian

I want to calculate the Kullbach Leibler Divergence of two multivariate Gaussians as in KL divergence between two multivariate Gaussians. At one point one has to solve the following expression (...
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157 views

KL divergence between sample and true (multivariate normal) distribution

I was wondering, whether there is a possible interpretation of the KL-Divergence between sample and true distribution in terms of probabilities. E.g. given $P=\mathcal{N}\left(\mu,\Sigma\right)$ and $...
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207 views

Gaussian-to-gaussian transformations

It is well-known that when a linear transformation is applied to a normally-distributed random variable, the result is itself a normally-distributed random variable. I am interested in the converse ...
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2answers
51 views

Using $r_Q$ statistic to check multivariate normality

Suppose in R, I have a data set data(iris) and I want to check for multivariate normality of the first four columns ("Sepal.Length" "Sepal.Width" "Petal.Length"...
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Is Cov(X,Y|Z).x always positive? (with X,Y,Z, normal random vectors and x>0)

Let x be a vector of positive values, we know that for multivariate normal distributions of X, Y and Z, $Cov(X,Y|Z)x=(\Sigma_{XZ}-\Sigma_{XY}\Sigma_{YY}^{-1}\Sigma_{YZ})x$ does not depend on the given ...
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two-sample $T^2$ testing with $n=1$

Assuming I have two populations $X\sim\mathcal{N}(\mu_X,\Sigma)$ and $Y\sim\mathcal{N}(\mu_Y,\Sigma)$ of sizes $n_X,n_Y$, the hypotheses for equivalence of means (each of length $p$, and assuming ...
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2answers
60 views

Normal distributed random variables with constraint?

Consider $n$ random variables $X_i$ with $i=1,2,...,n$, each drawing values from identical normal distributions with mean $\mu=0$ and standard deviation $\sigma=const.$ so that expectation values are $...
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331 views

Calculating P-value for multivariate normal distributions?

I'm given a set of 500 ($\sigma_i, \mu_i$) that define a 500-dimensional multivariate normal distribution, where each dimension is effectively independent from all the others. Given a 500-...