$\newcommand{\Var}{\operatorname{Var}}$$\newcommand{\Cov}{\operatorname{Cov}}$$\newcommand{\one}{\mathbf 1}$$\newcommand{\0}{\mathbf 0}$$\newcommand{\e}{\varepsilon}$$\newcommand{\E}{\operatorname{E}}$In this post I’m going to discuss leverage scores in linear regression. In particular, I’ll show how the…
$\newcommand{\e}{\varepsilon}$$\newcommand{\Var}{\operatorname{Var}}$$\newcommand{\x}{\mathbf x}$$\newcommand{\one}{\mathbf 1}$$\newcommand{\0}{\mathbf 0}$In this post I will derive the point estimates for mixed models. I’ll focus on interpretation so…
$\newcommand{\R}{\mathbb R}$$\newcommand{\y}{\mathbf y}$$\newcommand{\thb}{\tilde{\hat\beta}}$$\newcommand{\L}{\Lambda}$$\newcommand{\tr}{\operatorname{tr}}$In this post I’ll look at updating a linear regression with new data. One place where this can…
$\newcommand{\R}{\mathbb R}$$\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}$$\newcommand{\E}{\operatorname{E}}$Let $\mu$ be a probability measure on $(\mathbb R, \mathbb B)$ where $\mathbb B$ is the Borel $\sigma$-algebra…
$\newcommand{\e}{\varepsilon}$In this post I’ll be exploring non-integer bases and the “natural numbers” they represent. I say “natural numbers” because the…
In this post I’m going to prove some results needed to solve the “XOR-sequence” HackerRank problem found here. Spoiler warning…
$\newcommand{\logit}{\operatorname{logit}}$$\newcommand{\y}{\mathbf y}$$\newcommand{\one}{\mathbf 1}$$\newcommand{\e}{\varepsilon}$$\newcommand{\0}{\mathbf 0}$$\newcommand{\Var}{\operatorname{Var}}$$\newcommand{\E}{\operatorname{E}}$In this post I’ll look at the gradient of the loss in logistic regression and explore a…
$\newcommand{\one}{\mathbf 1}$$\newcommand{\0}{\mathbf 0}$$\newcommand{\tr}{\operatorname{tr}}$$\newcommand{\e}{\varepsilon}$In this post I’m going to look at some relationships between the diagonal of a positive semidefinite (PSD)…