brycewang-stanford/proof-architect

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