---
id: wiki-2026-0508-statistics
title: Statistics
category: 10_Wiki/Topics
status: verified
canonical_id: self
aliases: [Stats, Statistical-Inference]
duplicate_of: none
source_trust_level: A
confidence_score: 0.95
verification_status: applied
tags: [statistics, inference, frequentist, bayesian, foundations]
raw_sources: []
last_reinforced: 2026-05-10
github_commit: pending
tech_stack:
  language: python
  framework: scipy-statsmodels-pymc
---

# Statistics

## 매 한 줄
> **"매 data → 매 truth 의 inference"**. Statistics 의 uncertainty 하의 decision-making — descriptive (summary), inferential (population from sample), causal (intervention) 의 3 axes. 17C Pascal/Fermat 에서 시작, 2026 에 frequentist + Bayesian + ML 의 hybrid 의 표준.

## 매 핵심

### 매 3 axes
- **Descriptive**: mean, median, SD, distribution, viz. 매 "what is".
- **Inferential**: confidence interval, hypothesis test, p-value, posterior. 매 "what could be".
- **Causal**: RCT, propensity, IV, DAG. 매 "what if".

### 매 frequentist vs Bayesian
- **Frequentist**: probability = 매 long-run frequency. 매 parameters fixed, data random. p-value, CI.
- **Bayesian**: probability = 매 belief degree. 매 parameters random (prior + likelihood → posterior). credible interval.
- 매 ML community 의 mostly Bayesian-ish (regularization = prior).

### 매 inference workflow
1. Question → estimand 정의.
2. Design → sample, randomization, power.
3. Data collection.
4. Model — distribution / link function.
5. Estimate — MLE, MAP, posterior.
6. Inference — CI / credible interval, test.
7. Critique — diagnostics, sensitivity, replication.

### 매 응용
1. A/B testing — frequentist or Bayesian decision.
2. Survey / poll — sampling design.
3. Clinical trial — RCT, survival analysis.
4. ML model evaluation — bootstrap, cross-validation.
5. Causal inference — econometrics, epidemiology.

## 💻 패턴

### 1. Descriptive — pandas
```python
import pandas as pd
df.describe()              # mean, std, quartiles
df.groupby("group").agg(["mean", "std", "count"])
```

### 2. Hypothesis test — scipy
```python
from scipy import stats
t, p = stats.ttest_ind(a, b, equal_var=False)  # Welch's t
u, p = stats.mannwhitneyu(a, b)                # nonparametric
chi, p, dof, exp = stats.chi2_contingency(table)
```

### 3. Confidence interval (bootstrap, distribution-free)
```python
import numpy as np
def bootstrap_ci(x, stat=np.mean, B=10000, alpha=0.05):
    boot = [stat(np.random.choice(x, len(x), replace=True)) for _ in range(B)]
    return np.percentile(boot, [100*alpha/2, 100*(1-alpha/2)])
```

### 4. Linear regression with inference (statsmodels)
```python
import statsmodels.api as sm
X = sm.add_constant(df[["x1", "x2"]])
model = sm.OLS(df["y"], X).fit()
print(model.summary())  # coef, SE, p, R²
print(model.conf_int())
```

### 5. Bayesian inference (PyMC, 2026)
```python
import pymc as pm
with pm.Model() as m:
    mu = pm.Normal("mu", 0, 10)
    sigma = pm.HalfNormal("sigma", 5)
    y = pm.Normal("y", mu=mu, sigma=sigma, observed=data)
    idata = pm.sample(2000, tune=1000, chains=4, target_accept=0.95)
import arviz as az
az.summary(idata, var_names=["mu", "sigma"])
```

### 6. Multiple testing — BH (FDR)
```python
from statsmodels.stats.multitest import multipletests
reject, p_adj, _, _ = multipletests(p_values, alpha=0.05, method="fdr_bh")
```

### 7. Causal — propensity score matching
```python
from sklearn.linear_model import LogisticRegression
ps = LogisticRegression().fit(X, treatment).predict_proba(X)[:, 1]
# 매 nearest neighbor matching on logit(ps) → ATT estimate
```

### 8. Survival — Kaplan-Meier
```python
from lifelines import KaplanMeierFitter
kmf = KaplanMeierFitter()
kmf.fit(durations, event_observed=events)
kmf.plot_survival_function()
```

### 9. Cross-validation (proper inference for ML)
```python
from sklearn.model_selection import cross_val_score
scores = cross_val_score(model, X, y, cv=5, scoring="neg_mean_squared_error")
ci = (scores.mean() - 1.96*scores.std(), scores.mean() + 1.96*scores.std())
```

## 매 결정 기준
| 상황 | Approach |
|---|---|
| Quick descriptive | pandas .describe() |
| 2-group mean diff | Welch's t (default) / Mann-Whitney (non-Gaussian) |
| Proportion test | chi-square / Fisher's exact (small n) |
| Complex hierarchy | Mixed effects (statsmodels.MixedLM) / Bayesian (PyMC) |
| Decision under uncertainty | Bayesian posterior |
| Causal claim | RCT > IV > DiD > matching |
| Many tests | FDR (BH) / FWER (Bonferroni for safety-critical) |
| Distribution-free CI | Bootstrap |

**기본값**: descriptive 먼저 → assumption 의 check → frequentist 의 CI/test 의 default → critical decision 시 Bayesian 의 considerable.

## 🔗 Graph
- 부모: [[Probability Theory]]
- 변형: [[Probability Theory|Probability-Theory-Foundations]] · [[Sampling-Techniques]]
- 응용: [[Regression-Analysis-Foundations]] · [[Statistical-Power]] · [[Multivariate-Analysis]]
- Adjacent: [[Standard-Deviation-and-Variance]] · [[Entropy in Information Theory|Information Theory]]

## 🤖 LLM 활용
**언제**: choosing appropriate test, interpreting output, explaining p-value/CI to non-technical, drafting analysis plan.
**언제 X**: 매 numerical computation — scipy/statsmodels/PyMC 의 사용.

## ❌ 안티패턴
- **p < 0.05 = 매 truth**: 매 decision threshold 의 X, just evidence — replication, effect size, prior 의 고려.
- **Correlation = causation**: 매 흔한 fallacy — confounding, reverse causality, selection.
- **Ignoring assumptions**: t-test 의 normality, regression 의 linearity → 매 diagnostic 의 필수.
- **Cherry-picking subgroups**: 매 fishing → false positive 의 폭증.
- **Reporting only mean (skewed data)**: median, IQR 의 추가.
- **Sample size 의 unspecified**: 매 power 의 미상.

## 🧪 검증 / 중복
- Verified (Wasserman *All of Statistics*, Gelman *BDA3*, ASA p-value statement 2016).
- 신뢰도 A.

## 🕓 Changelog
| 날짜 | 변경 |
|---|---|
| 2026-05-08 | Phase 1 |
| 2026-05-10 | Manual cleanup — frequentist/Bayesian/causal axes + workflow. |