proof-architect

name	proof-architect
description	Structured methodology for constructing and verifying mathematical proofs in statistical research

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.

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Proof Structure Framework

Standard Proof Components

Every rigorous statistical proof should contain:

1. Claim Statement - Precise mathematical statement of what is being proved
2. Assumptions - All conditions required (clearly enumerated A1, A2, ...)
3. Notation - Define all symbols before use
4. Proof Body - Logical sequence of justified steps
5. 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}

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Proof Types in Statistical Methodology

1. Identification Proofs

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

Standard Structure:

1. Define target estimand (e.g., $\psi = E[Y(a)]$)
2. State identifying assumptions (consistency, positivity, exchangeability)
3. Apply identification formula derivation
4. 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:

1. Define estimator $\hat{\theta}_n$
2. Define target parameter $\theta_0$
3. 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:

1. Taylor expansion around true value
2. Apply CLT to score/influence function
3. Invert Hessian/information matrix
4. 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:

1. Characterize the tangent space
2. Derive efficient influence function (EIF)
3. Show estimator's influence function equals EIF
4. 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:

1. Write estimating equation with both nuisance functions
2. Show bias term is product of two errors
3. 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}

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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?

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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

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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

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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

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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

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

Key References

• van der Vaart
• Lehmann
• Casella
• Bickel
• Serfling