Advanced Statistical Theory Maximum Likelihood Estimation MLE and Expectation-maximization EM

Advanced Statistical Theory Maximum Likelihood Estimation MLE and Expectation-maximization EM is part of the statistical bedrock that every model sits on top of. Understand it well and you stop treating p-values, confidence intervals and distributions as magic numbers and start reasoning about them as the tools they really are.

Why Advanced Statistical Theory Matters

Misusing statistical tools is how otherwise-talented teams ship confident but wrong conclusions. Solid foundations here protect you from hallucinated effects, under-powered studies and false certainty.

• Define the random variables and their distributions precisely.
• Choose estimators whose bias and variance you can reason about.
• Quantify uncertainty with confidence or credible intervals.
• Use p-values only as one piece of evidence, never the conclusion.

How Advanced Statistical Theory Shows Up in Practice

In a typical project, advanced statistical theory maximum likelihood estimation mle and expectation-maximization em is combined with the rest of the Statistics & Probability toolkit. You rarely use any one technique in isolation; the real skill is knowing which combination fits the problem you are trying to solve, and being able to explain that choice to a non-technical stakeholder.

Apply this material in any experiment, A/B test, survey analysis or report that will be used to make a real-world decision.

• Epistemology Data Scientists Paradigms Knowledge Inference
• Advanced Applied Statistics for Data Scientists
• Descriptive Statistics and Data Distribution Theory
• Principles of Statistical Inference and Estimation

Back to the Data Science curriculum →

# Example 1: Two-sample Mann-Whitney U test -- Advanced Statistical Theory Maximum Likelihood Estimation
import numpy as np
from scipy import stats

rng = np.random.default_rng(0)
a   = rng.gamma(shape=2.0, scale=1.0, size=120)    # right-skewed
b   = rng.gamma(shape=2.0, scale=1.2, size=140)

u_stat, p = stats.mannwhitneyu(a, b, alternative="two-sided")
effect    = 1 - 2 * u_stat / (len(a) * len(b))     # rank-biserial r

print(f"medians       : {np.median(a):.2f} vs {np.median(b):.2f}")
print(f"U statistic   : {u_stat:.0f}")
print(f"p-value       : {p:.4f}")
print(f"effect size r : {effect:+.3f}")

# Example 2: Chi-squared test of independence -- Advanced Statistical Theory Maximum Likelihood Estimation
import numpy as np
import pandas as pd
from scipy.stats import chi2_contingency

# Observed: plan type vs. churn outcome
observed = pd.DataFrame(
    [[180,  20],
     [220,  80],
     [150, 150]],
    index=["basic", "standard", "premium"],
    columns=["retained", "churned"],
)

chi2, p, dof, expected = chi2_contingency(observed.values)
cramer_v = np.sqrt(chi2 / (observed.values.sum() *
                            (min(observed.shape) - 1)))

print(observed)
print(f"\nchi2 = {chi2:.2f}  dof = {dof}  p = {p:.4g}")
print(f"Cramer's V = {cramer_v:.3f}")

# Example 3: One-sample t-test with 95% CI -- Advanced Statistical Theory Maximum Likelihood Estimation
import numpy as np
from scipy import stats

rng    = np.random.default_rng(7)
sample = rng.normal(loc=102.4, scale=14.0, size=60)
mu0    = 100.0

t_stat, p_val = stats.ttest_1samp(sample, popmean=mu0)
ci_lo, ci_hi  = stats.t.interval(0.95, df=len(sample) - 1,
                                 loc=sample.mean(),
                                 scale=stats.sem(sample))

print(f"mean    : {sample.mean():.2f}")
print(f"95% CI  : ({ci_lo:.2f}, {ci_hi:.2f})")
print(f"t, p    : {t_stat:.3f}, {p_val:.4f}")
print("verdict :", "reject H0" if p_val < 0.05 else "fail to reject H0")

# Example 4: Bayesian Beta-Binomial update -- Advanced Statistical Theory Maximum Likelihood Estimation
import numpy as np
from scipy import stats

prior_a, prior_b   = 2, 2          # weak Beta(2,2) prior
successes, trials  = 47, 80        # observed data

post_a = prior_a + successes
post_b = prior_b + (trials - successes)
post   = stats.beta(post_a, post_b)

print(f"posterior mean       : {post.mean():.3f}")
print(f"95% credible interval: "
      f"({post.ppf(0.025):.3f}, {post.ppf(0.975):.3f})")
print(f"P(p > 0.5 | data)    : {1 - post.cdf(0.5):.3f}")

samples = post.rvs(size=20_000, random_state=0)
print(f"Monte-Carlo check    : {samples.mean():.3f}")

# Example 5: Bootstrap CI for a robust statistic -- Advanced Statistical Theory Maximum Likelihood Estimation
import numpy as np

rng  = np.random.default_rng(1)
data = rng.lognormal(mean=1.2, sigma=0.4, size=200)

def bootstrap_ci(x, stat=np.median, B=5_000, alpha=0.05):
    n = len(x)
    draws = np.empty(B)
    for b in range(B):
        idx       = rng.integers(0, n, n)
        draws[b]  = stat(x[idx])
    lo, hi = np.quantile(draws, [alpha / 2, 1 - alpha / 2])
    return stat(x), lo, hi

point, lo, hi = bootstrap_ci(data, np.median)
print(f"median  = {point:.3f}  (95% CI: {lo:.3f}, {hi:.3f})")