---
id: wiki-2026-0508-inferential-statistics
title: Inferential Statistics
category: 10_Wiki/Topics
status: verified
canonical_id: self
aliases: [Statistical Inference, Hypothesis Testing, Confidence Intervals]
duplicate_of: none
source_trust_level: A
confidence_score: 0.92
verification_status: applied
tags: [statistics, inference, hypothesis-testing, ab-testing, sre]
raw_sources: []
last_reinforced: 2026-05-10
github_commit: pending
tech_stack:
  language: python
  framework: scipy
---

# Inferential Statistics

## 매 한 줄
> **"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.

## 매 핵심

### 매 Frequentist vs Bayesian
- **Frequentist**: parameter fixed, data random. p-value, CI.
- **Bayesian**: parameter random (prior), data fixed. Posterior, credible interval.
- **Bootstrap**: distribution-free, resample n→inf 시뮬레이션.

### 매 Test 분류
- **Parametric**: t-test, ANOVA, Z-test (assumes normal).
- **Non-parametric**: Mann-Whitney U, Kruskal-Wallis, permutation.
- **Sequential**: Always Valid Inference, mSPRT (peek-safe).

### 매 응용
1. A/B test: conversion lift 측정.
2. SRE: SLO breach 의 statistical significance.
3. ML: model A vs B 의 holdout 비교.

## 💻 패턴

### Two-sample t-test
```python
import scipy.stats as st
control = [12, 14, 11, 13, 12, 15, 13]
treat   = [16, 18, 15, 17, 19, 16, 18]
res = st.ttest_ind(control, treat, equal_var=False)
print(f"t={res.statistic:.3f} p={res.pvalue:.4f}")
ci = res.confidence_interval(0.95)
print(f"95% CI: [{ci.low:.2f}, {ci.high:.2f}]")
```

### Bootstrap CI
```python
import numpy as np
def bootstrap_mean_ci(x, n=10_000, alpha=0.05):
    rng = np.random.default_rng(42)
    boots = rng.choice(x, size=(n, len(x)), replace=True).mean(axis=1)
    return np.quantile(boots, [alpha/2, 1-alpha/2])

ci = bootstrap_mean_ci(np.array(control))
print(f"Bootstrap 95% CI: {ci}")
```

### Sample size calculation (power)
```python
from statsmodels.stats.power import TTestIndPower
analysis = TTestIndPower()
n = analysis.solve_power(effect_size=0.3, power=0.8, alpha=0.05)
print(f"매 group 당 n = {int(np.ceil(n))}")
```

### Sequential test (mSPRT, peek-safe)
```python
import numpy as np
def msprt_log_likelihood(x, mu0=0, sigma=1, theta=0.1):
    n = len(x); xbar = np.mean(x); v = sigma**2
    tau2 = theta**2
    log_bf = 0.5*np.log(v/(v+n*tau2)) + (n**2 * (xbar-mu0)**2 * tau2) / (2*v*(v+n*tau2))
    return log_bf  # > log(1/alpha) 매 reject H0
```

### Bayesian A/B (PyMC)
```python
import pymc as pm
with pm.Model() as m:
    p_a = pm.Beta("p_a", 1, 1)
    p_b = pm.Beta("p_b", 1, 1)
    pm.Binomial("y_a", n=10_000, p=p_a, observed=520)
    pm.Binomial("y_b", n=10_000, p=p_b, observed=580)
    diff = pm.Deterministic("diff", p_b - p_a)
    idata = pm.sample(2000, chains=4, random_seed=42)
print(f"P(B > A) = {(idata.posterior['diff'] > 0).mean().item():.3f}")
```

### Permutation test
```python
def permutation_test(a, b, n=10_000):
    diff_obs = np.mean(a) - np.mean(b)
    pool = np.concatenate([a, b])
    rng = np.random.default_rng(0)
    diffs = []
    for _ in range(n):
        rng.shuffle(pool)
        diffs.append(np.mean(pool[:len(a)]) - np.mean(pool[len(a):]))
    return np.mean(np.abs(diffs) >= abs(diff_obs))
```

### SRE: Welch's test on latency p99
```python
# 매 deploy 전후 latency p99 비교
from scipy.stats import ttest_ind
before_p99 = np.array([124, 130, 128, 132, 125])  # ms
after_p99  = np.array([142, 138, 145, 140, 144])
t, p = ttest_ind(before_p99, after_p99, equal_var=False)
if p < 0.01: print("매 regression detected — rollback")
```

## 매 결정 기준
| 상황 | Approach |
|---|---|
| Fixed-N A/B | t-test or chi-squared |
| Continuous monitoring | mSPRT or always-valid CI |
| Small N, non-normal | Bootstrap or permutation |
| Multi-arm + prior | Bayesian (Beta-Binomial) |

**기본값**: Bootstrap CI + sequential test 의 production A/B.

## 🔗 Graph
- 부모: [[Statistics & Data Analysis]] · [[Probability Theory]]
- 변형: [[Bayesian_Inference|Bayesian Inference]]
- 응용: [[SRE]] · [[Anomaly-Detection]]
- Adjacent: [[Type 1 vs Type 2 Errors]] · [[Power Analysis]]

## 🤖 LLM 활용
**언제**: test 선택 의 advice (data shape → test type), 의 result interpretation.
**언제 X**: 매 multiple-comparison correction 매 자동화 X — domain knowledge 필요.

## ❌ 안티패턴
- **p-hacking**: 매 multiple test 후 cherry-pick.
- **Peeking**: fixed-N test 의 매 day 확인 → α inflation.
- **Single point**: CI 매 보고 안하고 mean 만.
- **N=∞ → significance ≠ effect size**: Cohen's d 도 같이.

## 🧪 검증 / 중복
- Verified (Casella & Berger "Statistical Inference", scipy/statsmodels docs).
- 신뢰도 A.

## 🕓 Changelog
| 날짜 | 변경 |
|---|---|
| 2026-05-08 | Phase 1 |
| 2026-05-10 | Manual cleanup — frequentist + Bayesian + sequential pattern |