Tagged Questions

1
vote
0answers
5 views

Confusion related to dual problem formulation in sparse inverse covariance matrix estimation

I was reading this paper where they are trying to estimate the inverse covariance matrix of the gaussian. What they are trying to maximize the gaussian log likelihood. The primal problem is maximize ...
4
votes
1answer
63 views

Given samples from multiple normal RVs, how do we recover the histogram of their means?

Let $X_1,...,X_N$ be independent normal random variables. $X_i$ is normal with mean $\mu_i$ and standard deviation $\sigma_i$. Let $x_i$ be a single random sample from $X_i$. Input: We get all ...
0
votes
0answers
31 views

pdf of multivariate normal distribution

I have a question concerning some sentences in the book Structural Equations with Latent Variables (Bollen) at page 132 (bottom) and page 133 (top) regarding the pdf of the multivariate normal ...
1
vote
1answer
225 views

Hypothesis testing of normal distribution, known mean unknown variance

I've been working on review problems, and this one has me completely stumped. Let $X_1 ... X_{10}$ be a random sample from a $N(3,\sigma^2)$ distribution, where $\sigma^2$ is unknown. Using the ...
0
votes
1answer
88 views

Spherical Gaussian Sigma dimension

I think I am confused with this thing. If we have a 3 dimension Gaussian then the MLE estimate for $\mu$ is a vector with 3 element $$\mu(1)' = \frac{1}{n}\sum_{j = 1} ^ n x_j\text{ and so on ...
0
votes
0answers
58 views

Constraints on ML for mixture of Gaussians

I have some data sampled from a mixture of two Gaussians where one of them is known, and the density function is as follows: $f(x, \mu, \sigma) = \frac{1}{2}\frac{1}{\sqrt{2\pi}\sigma} ...
3
votes
1answer
286 views

Maximum likelihood of function of the mean on a restricted parameter space

I've been trying to teach myself some of the fundamentals of statistics by trying to work through old qualifying exams. Here's a problem: Suppose $X_1, \ldots, X_n$ are a random sample from a normal ...
2
votes
1answer
112 views

Confusion related to derivation of the log likelihood

I was reading the Hastie, Friedman, Tibshirani paper "Sparse Inverse Covariance Estimation with the Graphical Lasso" and it had the following I couldn't get how the following expression was derived ...
9
votes
1answer
242 views

Hypothesis testing on the inverse covariance matrix

Suppose I observe i.i.d. $x_i \sim \mathcal{N}\left(\mu,\Sigma\right)$, and wish to test $H_0: A\ $vech$\left(\Sigma^{-1}\right) = a$ for a conformable matrix $A$ and vector $a$. Is there known work ...
0
votes
1answer
100 views

How can I replace this condition by a probability?

I want to see if a datapoint x should (or not) be assigned to a nearest component y using the following condition: if ($d > T$) then {do not assign x to y}. With $d = distance(x,y)$ and $T = ...
1
vote
0answers
187 views

Is there a covariance MLE which takes into account independence relationships?

In the extreme case where all of the components of an $M$-variate observation are pairwise independent from each other, a multivariate normal distribution can be decomposed into the product of $M$ ...
1
vote
2answers
405 views

How difficult is it to train a gaussian mixture model compared to other models?

I have finally been able to wrap my head around the mechanics of how to initialize and train a multivariate Gaussian mixture model using expectation maximization algorithm. So I wonder how difficult ...
1
vote
1answer
515 views

Likelihood at MLE and transformations, the multivariate normal case

Given a univariate sample $\vec X = X_1, ..., X_n$ with standard deviation 1 and a strictly monotone transformation $t: R \to R$ with the property that the standard deviation of $t(\vec X)$ is also 1 ...
1
vote
1answer
716 views

What is the maximum likelihood estimator of the mean of two normally-distributed variables?

Let $Z=(X+Y)/2$, where $X$ and $Y$ are independent normally-distributed random variables with known variances $\sigma^2_X$ and $\sigma^2_Y$ and unknown (and possibly different) means. Given a sample ...