Paul Hirschi
I build models that turn noisy data into decisions — and I write about the math that makes them trustworthy.
I'm a data scientist working across applied mathematics and statistics, from Bayesian methods and experimentation to the numerical guts of the algorithms everyone reaches for. Most of my work lives where a clean derivation meets a messy dataset.
This site is where I think out loud: posts that work through the math end to end, and a portfolio of engagements where that math moved a real number. If something here is useful, wrong, or worth arguing about, get in touch.
KEY EQUATIONS · APPLIED MATH
P(A \mid B) = \dfrac{P(B \mid A)\,P(A)}{P(B)}
Bayes’ Theorem
Inverts conditional probability to update beliefs as evidence arrives — the foundation of statistical inference.
\int_a^b f'(x)\,dx = f(b) - f(a)
Fundamental Theorem of Calculus
Ties differentiation and integration together, turning accumulated rates of change into exact totals.
f(x) = \dfrac{1}{\sigma\sqrt{2\pi}}\,\exp\!\left(-\dfrac{(x-\mu)^2}{2\sigma^2}\right)
Normal Distribution
The bell curve that sample averages converge to — the default model for noise and measurement error.
\sqrt{n}\,\big(\bar{X}_n - \mu\big) \;\xrightarrow{\,d\,}\; \mathcal{N}\!\big(0,\,\sigma^2\big)
Central Limit Theorem
Sample means look Gaussian no matter the population — the license behind most of applied inference.
\hat{\boldsymbol{\beta}} = \big(X^{\top}X\big)^{-1} X^{\top} y
Ordinary Least Squares
The closed-form solution to linear regression — the workhorse that fits a line through data.
\theta_{t+1} = \theta_t - \eta\,\nabla_{\!\theta}\,\mathcal{L}(\theta_t)
Gradient Descent
Steps downhill along the gradient to minimize loss — the optimizer behind nearly all machine learning.
\hat{\theta} = \operatorname*{arg\,max}_{\theta}\ \prod_{i=1}^{n} p(x_i \mid \theta)
Maximum Likelihood Estimation
Picks the parameters that make the observed data most probable — the dominant principle for fitting models.
H(X) = -\sum_{i} p_i \log p_i
Shannon Entropy
Measures the uncertainty in a distribution — the basis of information theory, coding, and decision trees.
f(x) = \sum_{n=0}^{\infty} \dfrac{f^{(n)}(a)}{n!}\,(x-a)^n
Taylor Series
Approximates any smooth function by polynomials — the engine of numerical methods and local analysis.
e^{i\pi} + 1 = 0
Euler’s Identity
Unites e, i, π, 1, and 0 in a single line — calculus’s gateway to Fourier analysis.