2nd/10_Wiki/Topics/Computer_Science_and_Theory/Statistics.md

id, title, category, status, canonical_id, aliases, duplicate_of, source_trust_level, confidence_score, verification_status, tags, raw_sources, last_reinforced, github_commit, tech_stack

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

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

2. Hypothesis test — scipy

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)

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)

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)

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)

from statsmodels.stats.multitest import multipletests
reject, p_adj, _, _ = multipletests(p_values, alpha=0.05, method="fdr_bh")

7. Causal — propensity score matching

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

from lifelines import KaplanMeierFitter
kmf = KaplanMeierFitter()
kmf.fit(durations, event_observed=events)
kmf.plot_survival_function()

9. Cross-validation (proper inference for ML)

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 · Mathematics
• 변형: Probability-Theory-Foundations · Sampling-Techniques
• 응용: Regression-Analysis-Foundations · Statistical-Power · Multivariate-Analysis
• Adjacent: Standard-Deviation-and-Variance · 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.