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$\newcommand{\Si}{\Sigma}$ It does not matter whether the random variable (r.v.) $R:=\Phi$ is discrete or continuous or neither; it can be any r.v. whatsoever, with values in any measurable space $(S, …

One can measure the rate of convergence of the posterior distribution with density $p_n$ to the Dirac probability distribution at $\theta_0$ by how large the ratios $p_n(\theta_0)/p_n(\theta)$ are for …

The pushforward measure of a Gaussian distribution in $\mathbb R^{n+1}$ under the map $\mathbb R^{n+1}\ni(x_0,x_1,\ldots,x_n)\mapsto(e^{x_0},\ldots,e^{x_n})$ is called a multivariate lognormal distrib …

The probability -- say $p$ -- for the experiment of rolling two different colored dice is $$\frac{162601421574468954588}{2^{2\times36}}\approx0.0344322.$$ Here it is assumed that the random sets $A$ a …

The pdf of the sum $X+Y$ of independent random variables $X$ and $Y$ is the convolution of the pdf's of $X$ and $Y$. The convolution operation is bilinear. The convolution of Gaussian pdf's is Gaussia …

$\newcommand\Si\Sigma\newcommand\X{\mathbf X}$If $Y,X_1,\dots,X_n$ are jointly normal zero-mean (real-valued) random variables, then $$E(Y|X_1,\dots,X_n)=\Si_{12}\Si_{22}^{-1}\X,$$ where $\X:=[X_1,\do …

$\newcommand\B{\mathscr B}\newcommand\C{\mathscr C}$There is hardly any particular intuition behind the concept of the cylindrical $\sigma$-algebra. This is just the smallest $\sigma$-algebra with res …

This has nothing to do with GLMM's per se. All what is done here is using the definition $$f_{Y|X}(y|x):=\frac{f_{X,Y}(x,y)}{f_X(x)}$$ (if $f_X(x)\ne0$) to write $$f_{Y|X}(y|x)f_X(x)=f_{X,Y}(x,y),$$ …

$\newcommand\th\theta$Given a family of pdf's $p_\th$ and a prior pdf $g$, the maximizer of $$F(q,g)(y):=E_q\ln p_\th(y)-D_{KL}(q\parallel g)$$ is always unique -- if any two pdf's differing only on a …