Project 6: Mathematical Foundations Proving Ground

Build a proof-and-computation workbook that validates the mathematical contracts behind practical statistics.

Quick Reference

Attribute	Value
Difficulty	Level 2: Intermediate
Time Estimate	1 week
Main Programming Language	Python (pseudocode-first)
Alternative Programming Languages	R, Julia
Coolness Level	Level 3: Genuinely Clever
Business Potential	1. The “Resume Gold”
Prerequisites	Algebra, basic matrices, probability basics
Key Topics	Sets, logic, sigma notation, logs, light calculus, linear algebra

1. Learning Objectives

By completing this project, you will:

Translate set/logic rules into event expressions used in statistics.
Use sigma notation and algebraic transforms to interpret estimator formulas.
Diagnose rank deficiency and invertibility issues in design matrices.
Explain how eigen concepts connect to variance decomposition and PCA.

2. All Theory Needed (Per-Concept Breakdown)

2.1 Set and Logic Foundations

Fundamentals: Sets and logic define valid event operations and prevent invalid inferential moves.
Deep Dive into the concept: Union/intersection/complement rules become probability rules once a measure is assigned. Logical equivalences (DeMorgan, implication forms) map directly into event transformations and conditional statements.
How this fit on projects: Supports event definitions in probability and inference projects.

Minimal concrete example:

Pseudo:
A = users_with_purchase
B = users_with_coupon
P(A|B) depends on A intersect B and B

2.2 Linear Algebra for Statistics

Fundamentals: Vectors/matrices encode datasets and model equations.
Deep Dive into the concept: Rank determines identifiability; near-singularity causes unstable coefficients; eigenvectors reveal dominant variation directions.

Mental model diagram:

Rows = observations, columns = features
X -> X^T X
if invertible: stable solve
if singular: non-unique coefficients

3. Project Specification

3.1 What You Will Build

A workbook-driven CLI tool that checks logic identities, verifies algebraic transformations, and audits matrix properties relevant to regression and PCA.

3.2 Functional Requirements

Validate core set/logic identities on random finite universes.
Verify sigma-form estimators against expanded algebraic forms.
Generate matrix rank/invertibility/condition diagnostics.
Produce an integrated markdown report with pass/fail and commentary.

3.3 Non-Functional Requirements

Performance: Complete < 30 seconds on moderate synthetic datasets.
Reliability: Deterministic outputs with fixed seeds.
Usability: Single command run with human-readable report.

3.4 Example Usage / Output

$ python foundations_workbook.py --profile stats
[PASS] Set identity tests: 200/200
[PASS] Sigma expansion equivalence: 24/24
[WARN] Rank deficiency found in scenario_03
[PASS] Eigen reconstruction error < 1e-06
Report saved: outputs/foundations_report.md

3.5 Real World Outcome

You will have a reusable “math sanity” report that catches design-matrix and formula mistakes before they contaminate inference or modeling work.

4. Solution Architecture

4.1 High-Level Design

Scenario generator -> identity/matrix checks -> diagnostics aggregator -> markdown report

4.2 Key Components

Component	Responsibility	Key Decisions
Identity Checker	Validates set/logic and algebra equivalences	finite-universe random tests
Matrix Auditor	Rank, invertibility, condition checks	tolerances and thresholds
Report Builder	Summarizes checks and implications	compact but explicit narratives

5. Implementation Guide

5.1 Development Environment Setup

pip install numpy scipy

5.2 Project Structure

P06/
  foundations_workbook.py
  scenarios/
  outputs/
  README.md

5.3 The Core Question You Are Answering

“Do I understand the mathematical assumptions well enough to trust later statistical outputs?”

5.4 Concepts You Must Understand First

Set/event algebra
Log and exponential transforms
Matrix rank and invertibility
Eigen decomposition intuition

5.5 Questions to Guide Your Design

How will you define numeric equivalence tolerances?
Which warnings should block downstream projects?

5.6 Thinking Exercise

Sketch one failure chain from “rank deficiency” to “misleading coefficients” to “bad decision”.

5.7 The Interview Questions They’ll Ask

Why does singular X^T X matter?
Why use logs in likelihoods?
How do eigenvectors connect to PCA?
What is condition number signaling?
Why test identities computationally?

5.8 Hints in Layers

Hint 1: Start with tiny finite sets and small matrices.
Hint 2: Add randomized scenarios for coverage.
Hint 3: Compare symbolic and numeric forms.
Hint 4: Track failure signatures in report output.

5.9 Books That Will Help

Topic	Book	Chapter
Linear algebra	Strang	Ch. 1-3
Probability foundations	Blitzstein & Hwang	Ch. 1
Inference math contracts	Casella & Berger	Likelihood chapters

6. Testing Strategy

Unit tests for identity checks.
Synthetic matrix test suite (full rank, rank deficient, ill-conditioned).
Deterministic rerun test with fixed seed.

7. Common Pitfalls & Debugging

Pitfall	Symptom	Solution
Overly strict tolerances	many false failures	scale-aware tolerance rules
Ignoring condition numbers	unstable downstream coefficients	warn + gate on threshold
No seed control	non-reproducible reports	fixed seed and run manifest

8. Extensions & Challenges

Add symbolic algebra validation with CAS tools.
Add visualization of eigen directions on toy datasets.

9. Real-World Connections

Model risk review in finance.
Feature matrix audits in ML pipelines.

10. Resources

Strang, “Introduction to Linear Algebra”
Casella & Berger, “Statistical Inference”

11. Self-Assessment Checklist

I can explain rank deficiency in plain language.
I can map one algebraic transform to a modeling benefit.
My report is deterministic and reproducible.

12. Submission / Completion Criteria

Minimum Viable Completion: identity checks + matrix audit + one report.

Full Completion: includes failure implications and mitigation guidance.