Proof Architect

Structured methodology for constructing and verifying mathematical proofs in statistical research

Use this skill when working on: mathematical proofs, theorem development, derivations, consistency proofs, asymptotic arguments, identification proofs, or verifying proof correctness.

Proof Structure Framework

Standard Proof Components

Every rigorous statistical proof should contain:

• Claim Statement - Precise mathematical statement of what is being proved

• Assumptions - All conditions required (clearly enumerated A1, A2, ...)

• Notation - Define all symbols before use

• Proof Body - Logical sequence of justified steps

• Conclusion - Explicit statement that claim is established

Proof Skeleton Template

\begin{theorem}[Name] \label{thm:name} Under Assumptions \ref{A1}--\ref{An}, [precise claim]. \end{theorem}

\begin{proof} The proof proceeds in [n] steps.

\textbf{Step 1: [Description]} [Content with justification for each transition]

\textbf{Step 2: [Description]} [Content]

\vdots

\textbf{Step n: Conclusion} Combining Steps 1--[n-1], we obtain [result], completing the proof. \end{proof}

Proof Types in Statistical Methodology

1. Identification Proofs

Goal: Show that a causal/statistical quantity is uniquely determined from observed data distribution.

Standard Structure:

• Define target estimand (e.g., $\psi = E[Y(a)]$)

• State identifying assumptions (consistency, positivity, exchangeability)

• Apply identification formula derivation

• Show formula depends only on observable quantities

Template:

\begin{theorem}[Identification of $\psi$] Under Assumptions \ref{A:consistency}--\ref{A:positivity}, the causal effect $\psi = E[Y(a)]$ is identified by [ \psi = \int E[Y \mid A=a, X=x] , dP(x). ] \end{theorem}

\begin{proof} \begin{align} E[Y(a)] &= E[E[Y(a) \mid X]] && \text{(law of iterated expectations)} \ &= E[E[Y(a) \mid A=a, X]] && \text{(A\ref{A:exchangeability}: $Y(a) \indep A \mid X$)} \ &= E[E[Y \mid A=a, X]] && \text{(A\ref{A:consistency}: $Y = Y(A)$)} \ &= \int E[Y \mid A=a, X=x] , dP(x) && \text{(definition)} \end{align} which depends only on the observed data distribution. \end{proof}

2. Consistency Proofs

Goal: Show that an estimator converges to the true parameter value.

Standard Structure:

• Define estimator $\hat{\theta}_n$

• Define target parameter $\theta_0$

• Establish convergence: $\hat{\theta}_n \xrightarrow{p} \theta_0$

Key Tools:

• Law of Large Numbers (LLN)

• Continuous Mapping Theorem

• Slutsky's Theorem

• M-estimation theory

Template:

\begin{theorem}[Consistency] Under Assumptions \ref{A1}--\ref{An}, $\hat{\theta}_n \xrightarrow{p} \theta_0$. \end{theorem}

\begin{proof} Define $M_n(\theta) = n^{-1} \sum_{i=1}^n m(O_i; \theta)$ and $M(\theta) = E[m(O; \theta)]$.

\textbf{Step 1: Uniform convergence} By [ULLN conditions], $\sup_{\theta \in \Theta} |M_n(\theta) - M(\theta)| \xrightarrow{p} 0$.

\textbf{Step 2: Unique maximum} $M(\theta)$ is uniquely maximized at $\theta_0$ (by identifiability).

\textbf{Step 3: Conclusion} By standard M-estimation theory, Steps 1--2 imply $\hat{\theta}_n \xrightarrow{p} \theta_0$. \end{proof}

3. Asymptotic Normality Proofs

Goal: Establish $\sqrt{n}(\hat{\theta}_n - \theta_0) \xrightarrow{d} N(0, V)$.

Standard Structure:

• Taylor expansion around true value

• Apply CLT to score/influence function

• Invert Hessian/information matrix

• State limiting distribution

Key Tools:

• Central Limit Theorem (CLT)

• Delta Method

• Influence Function Theory

• Semiparametric Efficiency Theory

Template:

\begin{theorem}[Asymptotic Normality] Under Assumptions \ref{A1}--\ref{An}, [ \sqrt{n}(\hat{\theta}_n - \theta_0) \xrightarrow{d} N(0, V) ] where $V = E[\phi(O)\phi(O)^\top]$ and $\phi$ is the influence function. \end{theorem}

\begin{proof} \textbf{Step 1: Score equation} $\hat{\theta}_n$ solves $\mathbb{P}n[\psi(O; \theta)] = 0$ where $\psi = \partial\theta m$.

\textbf{Step 2: Taylor expansion} [ 0 = \mathbb{P}_n[\psi(O; \hat{\theta}_n)] = \mathbb{P}_n[\psi(O; \theta_0)] + \mathbb{P}_n[\dot{\psi}(O; \tilde{\theta})](\hat{\theta}_n - \theta_0) ]

\textbf{Step 3: Rearrangement} [ \sqrt{n}(\hat{\theta}_n - \theta_0) = -\left(\mathbb{P}_n[\dot{\psi}]\right)^{-1} \sqrt{n} \mathbb{P}_n[\psi(O; \theta_0)] ]

\textbf{Step 4: Apply CLT} $\sqrt{n} \mathbb{P}_n[\psi(O; \theta_0)] \xrightarrow{d} N(0, \text{Var}(\psi))$ by CLT.

\textbf{Step 5: Slutsky} $\mathbb{P}_n[\dot{\psi}] \xrightarrow{p} E[\dot{\psi}]$ by WLLN. Apply Slutsky's theorem. \end{proof}

4. Efficiency Proofs

Goal: Show estimator achieves semiparametric efficiency bound.

Standard Structure:

• Characterize the tangent space

• Derive efficient influence function (EIF)

• Show estimator's influence function equals EIF

• Conclude variance achieves bound

Template:

\begin{theorem}[Semiparametric Efficiency] $\hat{\theta}n$ is semiparametrically efficient with influence function [ \phi(O) = [optimal formula] ] achieving the efficiency bound $V{\text{eff}} = E[\phi(O)^2]$. \end{theorem}

5. Double Robustness Proofs

Goal: Show estimator is consistent if either nuisance model is correctly specified.

Standard Structure:

• Write estimating equation with both nuisance functions

• Show bias term is product of two errors

• Conclude: if either error is zero, estimator is consistent

Template:

\begin{theorem}[Double Robustness] The estimator $\hat{\psi}_{DR}$ is consistent if either: \begin{enumerate} \item The outcome model $\mu(a,x) = E[Y \mid A=a, X=x]$ is correctly specified, or \item The propensity score $\pi(x) = P(A=1 \mid X=x)$ is correctly specified. \end{enumerate} \end{theorem}

\begin{proof} The estimating equation has the form: [ \psi - \hat{\psi}_{DR} = E\left[\frac{(A-\pi)(Y-\mu)}{\pi(1-\pi)}\right] + o_p(1) ] The bias term $(A-\pi)(Y-\mu)$ is zero in expectation if either: \begin{itemize} \item $E[A-\pi \mid X] = 0$ (propensity correctly specified), or \item $E[Y-\mu \mid A, X] = 0$ (outcome correctly specified). \end{itemize} \end{proof}

Proof Verification Checklist

Level 1: Structure Check

• Claim clearly stated with all conditions

• All notation defined before use

• Logical flow apparent (steps labeled)

• Each step has explicit justification

• Conclusion explicitly stated

Level 2: Step Validation

For each step, verify:

• Mathematical operation is valid

• Cited results apply (check conditions)

• Inequalities have correct direction

• Limits/integrals converge

• Dimensions/types match

Level 3: Edge Cases

• Boundary cases handled (n=1, p=0, etc.)

• Degenerate cases addressed

• Assumptions actually used (not vacuous)

• What happens at assumption boundaries?

Level 4: Consistency

• Result matches intuition

• Special cases recover known results

• Numerical verification possible?

• Consistent with simulation evidence?

Common Proof Errors

Technical Errors

Error Example Fix

Interchanging limits $\lim \sum \neq \sum \lim$ Verify DCT/MCT conditions

Division by zero $1/\pi(x)$ when $\pi(x)=0$ State positivity assumption

Incorrect conditioning $E[Y \mid A,X] \neq E[Y \mid X]$ Check independence structure

Wrong norm $|f|2$ vs $|f|\infty$ Verify which space

Missing measurability Random variable not measurable State measurability

Logical Errors

Error Example Fix

Circular reasoning Using result to prove itself Check logical dependency

Unstated assumption "Clearly, X holds" Make all assumptions explicit

Incorrect quantifier $\exists$ vs $\forall$ Be precise about scope

Missing case Not handling $\theta = 0$ Enumerate all cases

Statistical Errors

Error Example Fix

Confusing $\xrightarrow{p}$ and $\xrightarrow{d}$ Different convergence modes State which mode

Ignoring dependence Applying iid CLT to dependent data Check independence

Wrong variance Using population variance for sample Distinguish estimator/parameter

Notation Standards (VanderWeele Convention)

Causal Quantities

Symbol Meaning

$Y(a)$ Potential outcome under treatment $a$

$Y(a,m)$ Potential outcome under $A=a$, $M=m$

$M(a)$ Potential mediator under treatment $a$

$NDE$ Natural Direct Effect: $E[Y(1,M(0)) - Y(0,M(0))]$

$NIE$ Natural Indirect Effect: $E[Y(1,M(1)) - Y(1,M(0))]$

$TE$ Total Effect: $E[Y(1) - Y(0)] = NDE + NIE$

$P_M$ Proportion Mediated: $NIE/TE$

Statistical Quantities

Symbol Meaning

$\theta_0$ True parameter value

$\hat{\theta}_n$ Estimator based on $n$ observations

$\phi(O)$ Influence function

$\mathbb{P}_n$ Empirical measure

$\mathbb{G}_n$ Empirical process: $\sqrt{n}(\mathbb{P}_n - P)$

Convergence

Symbol Meaning

$\xrightarrow{p}$ Convergence in probability

$\xrightarrow{d}$ Convergence in distribution

$\xrightarrow{a.s.}$ Almost sure convergence

$O_p(1)$ Bounded in probability

$o_p(1)$ Converges to zero in probability

Proof Construction Workflow

Step 1: Understand the Goal

• What exactly needs to be proved?

• What type of proof is this? (identification, consistency, etc.)

• What are the key challenges?

Step 2: Gather Tools

• What theorems/lemmas are available?

• What regularity conditions will be needed?

• Are there similar proofs to reference?

Step 3: Outline Structure

• Break into logical steps

• Identify the key technical step

• Plan how to handle edge cases

Step 4: Write First Draft

• Fill in details for each step

• Be explicit about every transition

• Note where conditions are used

Step 5: Verify

• Run through verification checklist

• Check each step independently

• Test special cases

Step 6: Polish

• Improve notation consistency

• Add intuitive explanations

• Ensure assumptions are minimal

Integration with Other Skills

This skill works with:

• identification-theory - For causal identification proofs

• asymptotic-theory - For inference proofs

• methods-paper-writer - For presenting proofs in manuscripts

• proof-verifier - For systematic verification

Version: 1.0 Created: 2025-12-08 Domain: Mathematical Statistics, Causal Inference

Key References

• van der Vaart

• Lehmann

• Casella

• Bickel

• Serfling