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Antigravity Agent
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id: wiki-2026-0508-inferential-statistics
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title: Inferential Statistics
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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: [Statistical Inference, Hypothesis Testing, Confidence Intervals]
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duplicate_of: none
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source_trust_level: A
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confidence_score: 0.92
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verification_status: applied
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tags: [statistics, inference, hypothesis-testing, ab-testing, sre]
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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: scipy
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---
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# Inferential Statistics
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## 매 한 줄
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> **"매 sample → population parameter 의 추정 + uncertainty 의 quantify"**. 매 1900s Fisher, Neyman, Pearson 의 frequentist framework, 매 2026 A/B test, SRE alerting, ML evaluation 의 backbone — Bayesian + bootstrap 의 modern hybrid 가 default.
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## 매 핵심
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### 매 Frequentist vs Bayesian
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- **Frequentist**: parameter fixed, data random. p-value, CI.
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- **Bayesian**: parameter random (prior), data fixed. Posterior, credible interval.
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- **Bootstrap**: distribution-free, resample n→inf 시뮬레이션.
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### 매 Test 분류
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- **Parametric**: t-test, ANOVA, Z-test (assumes normal).
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- **Non-parametric**: Mann-Whitney U, Kruskal-Wallis, permutation.
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- **Sequential**: Always Valid Inference, mSPRT (peek-safe).
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### 매 응용
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1. A/B test: conversion lift 측정.
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2. SRE: SLO breach 의 statistical significance.
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3. ML: model A vs B 의 holdout 비교.
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## 💻 패턴
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### Two-sample t-test
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```python
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import scipy.stats as st
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control = [12, 14, 11, 13, 12, 15, 13]
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treat = [16, 18, 15, 17, 19, 16, 18]
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res = st.ttest_ind(control, treat, equal_var=False)
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print(f"t={res.statistic:.3f} p={res.pvalue:.4f}")
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ci = res.confidence_interval(0.95)
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print(f"95% CI: [{ci.low:.2f}, {ci.high:.2f}]")
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```
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### Bootstrap CI
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```python
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import numpy as np
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def bootstrap_mean_ci(x, n=10_000, alpha=0.05):
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rng = np.random.default_rng(42)
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boots = rng.choice(x, size=(n, len(x)), replace=True).mean(axis=1)
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return np.quantile(boots, [alpha/2, 1-alpha/2])
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ci = bootstrap_mean_ci(np.array(control))
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print(f"Bootstrap 95% CI: {ci}")
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```
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### Sample size calculation (power)
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```python
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from statsmodels.stats.power import TTestIndPower
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analysis = TTestIndPower()
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n = analysis.solve_power(effect_size=0.3, power=0.8, alpha=0.05)
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print(f"매 group 당 n = {int(np.ceil(n))}")
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```
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### Sequential test (mSPRT, peek-safe)
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```python
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import numpy as np
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def msprt_log_likelihood(x, mu0=0, sigma=1, theta=0.1):
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n = len(x); xbar = np.mean(x); v = sigma**2
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tau2 = theta**2
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log_bf = 0.5*np.log(v/(v+n*tau2)) + (n**2 * (xbar-mu0)**2 * tau2) / (2*v*(v+n*tau2))
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return log_bf # > log(1/alpha) 매 reject H0
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```
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### Bayesian A/B (PyMC)
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```python
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import pymc as pm
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with pm.Model() as m:
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p_a = pm.Beta("p_a", 1, 1)
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p_b = pm.Beta("p_b", 1, 1)
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pm.Binomial("y_a", n=10_000, p=p_a, observed=520)
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pm.Binomial("y_b", n=10_000, p=p_b, observed=580)
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diff = pm.Deterministic("diff", p_b - p_a)
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idata = pm.sample(2000, chains=4, random_seed=42)
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print(f"P(B > A) = {(idata.posterior['diff'] > 0).mean().item():.3f}")
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```
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### Permutation test
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```python
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def permutation_test(a, b, n=10_000):
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diff_obs = np.mean(a) - np.mean(b)
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pool = np.concatenate([a, b])
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rng = np.random.default_rng(0)
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diffs = []
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for _ in range(n):
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rng.shuffle(pool)
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diffs.append(np.mean(pool[:len(a)]) - np.mean(pool[len(a):]))
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return np.mean(np.abs(diffs) >= abs(diff_obs))
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```
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### SRE: Welch's test on latency p99
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```python
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# 매 deploy 전후 latency p99 비교
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from scipy.stats import ttest_ind
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before_p99 = np.array([124, 130, 128, 132, 125]) # ms
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after_p99 = np.array([142, 138, 145, 140, 144])
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t, p = ttest_ind(before_p99, after_p99, equal_var=False)
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if p < 0.01: print("매 regression detected — rollback")
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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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| Fixed-N A/B | t-test or chi-squared |
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| Continuous monitoring | mSPRT or always-valid CI |
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| Small N, non-normal | Bootstrap or permutation |
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| Multi-arm + prior | Bayesian (Beta-Binomial) |
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**기본값**: Bootstrap CI + sequential test 의 production A/B.
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## 🔗 Graph
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- 부모: [[Statistics & Data Analysis]] · [[Probability Theory]]
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- 변형: [[Bayesian_Inference|Bayesian Inference]]
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- 응용: [[SRE]] · [[Anomaly-Detection]]
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- Adjacent: [[Type 1 vs Type 2 Errors]] · [[Power Analysis]]
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## 🤖 LLM 활용
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**언제**: test 선택 의 advice (data shape → test type), 의 result interpretation.
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**언제 X**: 매 multiple-comparison correction 매 자동화 X — domain knowledge 필요.
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## ❌ 안티패턴
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- **p-hacking**: 매 multiple test 후 cherry-pick.
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- **Peeking**: fixed-N test 의 매 day 확인 → α inflation.
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- **Single point**: CI 매 보고 안하고 mean 만.
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- **N=∞ → significance ≠ effect size**: Cohen's d 도 같이.
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## 🧪 검증 / 중복
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- Verified (Casella & Berger "Statistical Inference", scipy/statsmodels docs).
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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 — frequentist + Bayesian + sequential pattern |
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