mathematical-foundations

npx skills add brycewang-stanford/Auto-Empirical-Research-Skills --skill mathematical-foundations --agent claude-code

Same command for any agent — swap --agent for codex, cursor, copilot.

# Mathematical Foundations **Core mathematical statistics theory for rigorous methodology development** Use this skill when working on: theoretical derivations requiring classical statistics, optimality arguments, exponential family manipulations, decision-theoretic comparisons, or foundational probability theory. --- ## Sufficiency ### Sufficiency Definition A statistic $T(X)$ is **sufficient** for parameter $\theta$ if the conditional distribution of $X$ given $T(X)$ does not depend on $\theta$: $$P(X = x \mid T(X) = t, \theta) = P(X = x \mid T(X) = t)$$ ### Factorization Theorem (Neyman-Fisher) $T(X)$ is sufficient for $\theta$ if and only if the likelihood can be factored as: $$f(x; \theta) = g(T(x), \theta) \cdot h(x)$$ where $g$ depends on $x$ only through $T(x)$, and $h$ does not depend on $\theta$. ### Minimal Sufficiency A sufficient statistic $T$ is **minimal sufficient** if for any other sufficient statistic $U$, there exists a function $g$ such that $T = g(U)$. **Criterion**: $T(x) = T(y)$ if and only if $\frac{f(x; \theta)}{f(y; \theta)}$ is constant in $\theta$. ### R Implementation ```r #' Check Sufficiency via Factorization #' #' @param likelihood Function returning