$\newcommand{\Om}{\Omega}\newcommand{\F}{\mathscr F}\newcommand{\one}{\mathbf 1}\newcommand{\R}{\mathbb R}\newcommand{\e}{\varepsilon}\newcommand{\E}{\operatorname{E}}\newcommand{\Var}{\operatorname{Var}}\newcommand{\convas}{\stackrel{\text{a.s.}}{\to}}\newcommand{\w}{\omega}\newcommand{\N}{\mathbb N}\newcommand{\convp}{\stackrel{\text{p}}{\to}}$In this post I’m going to introduce almost sure convergence for sequences of random variables, compare…
$\newcommand{\R}{\mathbb R}\newcommand{\N}{\mathbb N}$In this post I’m going to evaluate $$ \lim_{n\to\infty} (n!)^{1/n} $$ by connecting it to the harmonic series.…
$\newcommand{\R}{\mathbb R}$$\newcommand{\e}{\varepsilon}$$\newcommand{\H}{\mathcal H}$$\newcommand{\0}{\mathbf 0}$In this post I’m going to prove that the squared exponential kernel is positive definite. I’ll finish…
$\newcommand{\F}{\mathscr F}$$\newcommand{\R}{\mathbb R}$$\newcommand{\A}{\mathscr A}$$\newcommand{\G}{\mathcal G}$$\newcommand{\E}{\operatorname E}$$\newcommand{\dp}{\,\text dP}$$\newcommand{\1}{\mathbf 1}$Let $(\Omega, \F, P)$ be a probability space and let $X : \Omega…
$\newcommand{\vp}{\varphi}$$\newcommand{\dmux}{\,\text d\mu(x)}$$\newcommand{\dmu}{\,\text d\mu}$$\newcommand{\E}{\operatorname{E}}$Let $\mu$ be a probability measure on $(\mathbb R, \mathbb B)$ where $\mathbb B$ is the Borel $\sigma$-algebra…
$\newcommand{\logit}{\operatorname{logit}}$$\newcommand{\y}{\mathbf y}$$\newcommand{\one}{\mathbf 1}$$\newcommand{\e}{\varepsilon}$$\newcommand{\0}{\mathbf 0}$$\newcommand{\Var}{\operatorname{Var}}$$\newcommand{\E}{\operatorname{E}}$In this post I’ll look at the gradient of the loss in logistic regression and explore a…
$\newcommand{\X}{\mathcal X}$$\newcommand{\dl}{\,\text d\lambda}$$\newcommand{\dx}{\,\text dx}$$\newcommand{\one}{\mathbf 1}$I’m going to take a look at functional linear regression, where I’ll find the best linear…
$\newcommand{\bern}{\text{Bern}}$$\newcommand{\vp}{\varphi}$$\newcommand{\e}{\varepsilon}$$\newcommand{\P}{\mathcal P}$$\newcommand{\Cov}{\text{Cov}}$$\newcommand{\E}{\text{E}}$$\newcommand{\one}{\mathbf 1}$$\newcommand{\s}{\mathbf s}$$\newcommand{\Disc}{\text{Disc}}$In this post I’m going to explore correlations between finitely-supported discrete variables. Bernoulli case I’ll begin with…
$\newcommand{\hb}{\hat\beta}$$\newcommand{\hbl}{\hat\beta_\lambda}$$\newcommand{\tb}{\tilde \beta}$$\newcommand{\L}{\mathcal L}$$\newcommand{\l}{\lambda}$$\newcommand{\e}{\varepsilon}$$\newcommand{\0}{\mathbf 0}$$\newcommand{\Lam}{\Lambda}$$\newcommand{\g}{\gamma}$$\newcommand{\D}{\mathcal D}$$\newcommand{\ht}{\hat\theta}$$\newcommand{\a}{\alpha}$In this post I’m going to explore ridge regression as a constrained optimization, and I’ll do…
$\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…