$\newcommand{\I}{\mathcal I}$$\newcommand{\N}{\mathbb N}$$\newcommand{\Q}{\mathbb Q}$$\newcommand{\R}{\mathbb R}$In this post I’m going to explore partitioning the unit interval $\I := [0,1]$ into uncountably…
$\newcommand{\F}{\mathscr F}$$\newcommand{\a}{\alpha}$$\newcommand{\N}{\mathbb N}$$\newcommand{\w}{\omega}$$\newcommand{\E}{\mathcal E}$In this post I’m going to explore uncountable additivity of $\sigma$-finite measures a bit. Let $\mathcal M…
$\newcommand{\var}{\text{Var}}$$\newcommand{\cov}{\text{Cov}}$$\newcommand{\E}{\text{E}}$$\newcommand{\one}{\mathbf 1}$$\newcommand{\0}{\mathbf 0}$$\newcommand{\a}{\alpha}$$\newcommand{\l}{\lambda}$$\newcommand{\s}{\sigma}$$\newcommand{\e}{\varepsilon}$$\newcommand{\t}{\theta}$In this post I’m going to consider a linear model of the form $y = \mu\one + Z\a…
$\newcommand{\a}{\alpha}$$\newcommand{\L}{\mathcal L}$$\newcommand{\one}{\mathbf 1}$$\newcommand{\x}{\mathbf x}$$\newcommand{\0}{\mathbf 0}$Let $G = (V,E)$ be an undirected and unweighted graph on $n$ vertices. I’ll have $A\in\{0,1\}^{n\times…
$\newcommand{\one}{\mathbf 1}$In this post I’m going to consider the following situation: there is a latent unobserved continuous-state Markov chain $Z…
$\newcommand{\e}{\varepsilon}$$\newcommand{\E}{\text{E}}$$\newcommand{\C}{\mathcal C}$$\newcommand{\0}{\mathbf 0}$Consider the linear model $y = X\beta+\e$ with $\E\e = \0$, $\E \e\e^T = \sigma^2 I$, and $X\in\mathbb…
$\newcommand{\e}{\varepsilon}$Suppose I’ve got a linear model $y = X\beta + \e$ with $\e \sim \mathcal N(0, \sigma^2 I)$ and $X\in\mathbb…
$\newcommand{\E}{\text{E}}$$\newcommand{\Var}{\text{Var}}$$\newcommand{\Y}{\mathcal Y}$Here I’m going to introduce Monte Carlo maximum likelihood estimation (MCMLE) and I’ll use it to fit a logistic…
$\newcommand{\x}{\mathbf x}$ In this post I’m going to dust off the shell method from first semester calculus and use it…
$\newcommand{\N}{\mathbb N}$ In this post I’m going to use Euclidean division to do changes of base for nonnegative integers. Euclidean…