Updated: 27/10/25 Ref: “Foundations of Applied Mathematics, Vol 2” by Humpherys, Jarvis, Evans. Consider the optimization problem \[{ {\begin{aligned} \text{minimize} \quad &\, f(x) \\ \text{subject to} \quad &\, g _1 (x) \leq 0, \\ &\, \quad \vdots \\ &\, g _m (x) \leq 0 \end{aligned}} }\] where ${ f ,}$ ${ g _i }$s are co... Read more 26 Oct 2025 - 7 minute read
Ref: “Convex Optimization” by Boyd, Vandenberghe. “Convex Optimization” by Boyd. Lecture-2. Link to lecture: Lecture. [Dual cones] Note that cones induce inequalities. Link to Stackexchange post: Link. Consider ${ \mathbb{R} ^n . }$ Let ${ K \subseteq \mathbb{R} ^n }$ be a cone. Consider the dual space \[{ (\mathbb{R} ^n)^{\ast} := \l... Read more 23 Oct 2025 - 3 minute read
Updated: 21/10/25 Ref: “Introduction to Mathematical Statistics” by Hogg, McKean, Craig. “Statistics for Mathematicians” by Panaretos. [Hypothesis Testing] Consider a random variable ${ X }$ with density ${ f(x; \theta) , }$ ${ \theta \in \Omega . }$ Suppose we think ${ \theta \in \Omega _0 }$ or ${ \theta \in \Omega _1 , }$ where ${ \... Read more 20 Oct 2025 - 6 minute read
Consider the plane ${ \mathbb{R} ^2 . }$ Consider the origin ${ (0, 0) . }$ Consider the natural error generating process: The random error vector is ${ \mathcal{E} = (\varepsilon _1, \varepsilon _2) . }$ The components ${ \varepsilon _1, \varepsilon _2 }$ are independent and identically distributed. The distribution of ${ \mathcal{E} }$... Read more 08 Oct 2025 - 1 minute read
Ref: “Data Analysis for Social Scientists” by Duflo, Ellison. Lec-5. Link to the lecture: Link. Scikit-learn’s Density estimation documentation. Link to the page: Link. “Introduction to Mathematical Statistics” by Hogg, McKean, Craig. “Introduction to Nonparametric Estimation” by Tsybakov. “Smoothing methods in Statistics” by Simonoff... Read more 04 Oct 2025 - 2 minute read