refactor(topics): 멀티 에이전트용 지식 재편 — _Common(공통 기본기) + Domain_* 구조
에이전트 8종(대화형/프로그래머 C·S/디자이너/설계자/기획자/QA/PD/PM)에게 [공통 기본 능력 + 롤별 Specialty] 2층으로 지식을 주입하기 위한 재분류. 문서 내용·포맷은 무수정, 폴더 이동만 (6,372개 문서 수 보존 확인). - Topic_Programming → Domain_Programming (내부 구조 보존) - Topic_Graphic → Domain_Design - Topic_Business → Domain_Product - Topic_General → Domain_General - _Common 신설: Math(구 Topic_Math_Specialty), Reasoning(구 General/From_Thinking & Reasoning), Reasoning_Creativity(구 General/From_창의성), Communication(Poetic_Blog_Writing + From_writing) - 타 도메인의 From_* 폴더는 유지 (출처 표기일 뿐, 이미 도메인에 맞게 분류된 문서) - 빈 폴더 정리 (memory/procedures) - 에이전트→폴더 매핑은 workspace의 .astra/agent-knowledge-map.json (9개 에이전트) Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
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Antigravity Agent
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---
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id: wiki-2026-0508-variance-rules
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title: Variance Rules
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category: 10_Wiki/Topics
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status: verified
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canonical_id: self
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aliases: [Variance Algebra, Var Properties, Bienaymé Identity]
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duplicate_of: none
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source_trust_level: A
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confidence_score: 0.9
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verification_status: applied
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tags: [statistics, probability, math, identity]
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raw_sources: []
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last_reinforced: 2026-05-10
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github_commit: pending
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tech_stack:
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language: Python
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framework: NumPy
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---
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# Variance Rules
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## 매 한 줄
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> **"매 random variable 의 spread 의 algebra — Var(aX + b) = a²Var(X), 매 independence 매 sum 의 add"**. 1853 Bienaymé 의 sum-of-independent identity 부터 매 modern propagation-of-uncertainty, finance VaR, ML loss decomposition 까지 — 매 variance algebra 의 매 day-1 statistics 의 still 매 most-used identity.
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## 매 핵심
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### 매 core identities
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- **Definition**: $\mathrm{Var}(X) = \mathbb{E}[(X - \mathbb{E}[X])^2] = \mathbb{E}[X^2] - \mathbb{E}[X]^2$.
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- **Affine**: $\mathrm{Var}(aX + b) = a^2 \mathrm{Var}(X)$ — 매 constant $b$ 의 drop.
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- **Sum**: $\mathrm{Var}(X + Y) = \mathrm{Var}(X) + \mathrm{Var}(Y) + 2\,\mathrm{Cov}(X, Y)$.
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- **Independence (Bienaymé)**: $X \perp Y \Rightarrow \mathrm{Var}(X+Y) = \mathrm{Var}(X) + \mathrm{Var}(Y)$.
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- **Linear comb**: $\mathrm{Var}\!\left(\sum a_i X_i\right) = \sum a_i^2 \mathrm{Var}(X_i) + 2 \sum_{i<j} a_i a_j \mathrm{Cov}(X_i, X_j)$.
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- **Law of total variance**: $\mathrm{Var}(Y) = \mathbb{E}[\mathrm{Var}(Y|X)] + \mathrm{Var}(\mathbb{E}[Y|X])$.
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- **Sample variance bias correction**: $s^2 = \frac{1}{n-1}\sum (x_i - \bar{x})^2$ — 매 Bessel.
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### 매 propagation (delta method)
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- **Univariate**: $\mathrm{Var}(g(X)) \approx (g'(\mu))^2 \mathrm{Var}(X)$.
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- **Multivariate**: $\mathrm{Var}(g(\mathbf{X})) \approx \nabla g(\mu)^\top \Sigma \nabla g(\mu)$.
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### 매 응용
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1. Portfolio variance (Markowitz).
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2. Error propagation in physics measurement.
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3. ML bias-variance decomposition.
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4. A/B test sample-size (Welch).
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5. Kalman filter — covariance propagation.
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## 💻 패턴
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### Numerical sample variance — Welford (numerically stable)
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```python
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def welford_variance(stream):
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n = 0; mean = 0.0; M2 = 0.0
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for x in stream:
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n += 1
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delta = x - mean
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mean += delta / n
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M2 += delta * (x - mean) # 매 use updated mean
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return mean, M2 / (n - 1) if n > 1 else float('nan')
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```
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### Linear combination variance
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```python
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import numpy as np
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def linear_combo_var(weights, cov):
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# Var(w^T X) = w^T Σ w
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w = np.asarray(weights); cov = np.asarray(cov)
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return float(w @ cov @ w)
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```
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### Portfolio variance (Markowitz)
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```python
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def portfolio_var(weights, returns_matrix):
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cov = np.cov(returns_matrix, rowvar=False, ddof=1)
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return weights @ cov @ weights
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```
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### Delta-method propagation
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```python
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import numpy as np
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def delta_method(g, grad_g, mu, sigma):
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# mu: vector, sigma: covariance
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g_grad = np.asarray(grad_g(mu))
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return float(g_grad @ sigma @ g_grad)
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```
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### Law of total variance — verify by simulation
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```python
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import numpy as np
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rng = np.random.default_rng(0)
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N = 1_000_000
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X = rng.integers(0, 3, size=N) # 매 latent class
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mu_y = np.array([0.0, 1.0, 5.0])[X]
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Y = rng.normal(mu_y, scale=1.0)
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total = Y.var()
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inner = np.array([Y[X==k].var() for k in range(3)]).mean()
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outer = np.array([Y[X==k].mean() for k in range(3)]).var()
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print(total, inner + outer) # 매 ≈ equal
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```
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### Bias-variance decomposition (ML)
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```python
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def bias_variance(predictions, y_true):
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# predictions: (n_models, n_samples)
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mean_pred = predictions.mean(axis=0)
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bias_sq = ((mean_pred - y_true) ** 2).mean()
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var = predictions.var(axis=0).mean()
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noise_lb = 0.0 # 매 estimable 의 의 separately
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return bias_sq, var, noise_lb
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```
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### Welch's t-test variance handling
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```python
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from scipy import stats
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t, p = stats.ttest_ind(a, b, equal_var=False) # Welch
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# 매 unequal variance — Satterthwaite degrees of freedom
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```
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## 매 결정 기준
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| 상황 | Approach |
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|---|---|
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| Streaming variance | Welford (numerically stable) |
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| Independent sum | Bienaymé — sum the variances |
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| Correlated sum | Full covariance — $w^\top \Sigma w$ |
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| Nonlinear function $g(X)$ | Delta method (1st-order) — or Monte Carlo |
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| Hierarchical / mixture | Law of total variance 의 decompose |
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| ML overfitting diagnose | Bias-variance decomposition |
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| Sample variance | Bessel correction ($n-1$) |
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**기본값**: independence 의 confirm 후 Bienaymé. Doubt — Monte Carlo 의 verify.
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## 🔗 Graph
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- 부모: [[Probability Theory]]
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- 응용: [[Bias-Variance Tradeoff]]
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## 🤖 LLM 활용
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**언제**: identity recall, derivation hint, code skeleton (Welford, delta).
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**언제 X**: 매 specific paper 의 closed-form — derivation 의 cross-check.
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## ❌ 안티패턴
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- **Bienaymé 의 correlated variable 의 apply**: 매 covariance 의 forget — biased toward zero variance.
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- **Two-pass naive variance** ($\sum x_i^2 - (\sum x_i)^2/n$): 매 catastrophic cancellation — Welford 의 use.
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- **Sample variance with $n$**: 매 biased — Bessel ($n-1$).
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- **Affine 매 $b$ 의 add to variance**: 매 $b$ 의 drop, only $a^2$ matters.
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- **Delta method 의 high curvature 의 use**: 매 1st-order — large $\sigma$ 의 의 break, 의 Monte Carlo.
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## 🧪 검증 / 중복
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- Verified (Bienaymé 1853; Casella & Berger _Statistical Inference_ 2nd ed.; Welford 1962 _Technometrics_).
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- 신뢰도 A.
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## 🕓 Changelog
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| 날짜 | 변경 |
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|---|---|
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| 2026-05-08 | Phase 1 |
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| 2026-05-10 | Manual cleanup — variance algebra + Welford + delta method 정리 |
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