Inference reference
The four frameworks side by side
This page lays the course’s four inferential frameworks next to one another so you can see, at a glance, what each conditions on, what it claims, and how to read its output. The running illustration is the recurring synthetic reading-fluency study — a pass rate \(\theta\) estimated from \(x = 26\) successes in \(n = 40\) — so the same numbers recur across the lenses and the differences are differences of meaning, not of arithmetic. Every value here is synthetic and verified: false (the math gate is blocked pending sign-off).
The big picture
All four frameworks answer “what does the data say about the parameter \(\theta\)?” The deep difference is conditioning — what each treats as fixed and what it treats as random:
- Frequentist — \(\theta\) is fixed; the data are random. Claims are about the long-run behavior of procedures (coverage, error rates).
- Likelihood — the data are fixed; parameter values are ranked by how well they explain the data, with no probability placed over \(\theta\).
- Simulation-based — resample or relabel the data by computer to build a reference distribution, with minimal distributional assumptions.
- Bayesian — the data are fixed; \(\theta\) is uncertain and carries a probability distribution. Claims are posterior probabilities about \(\theta\).
Comparison table
| Frequentist | Likelihood | Simulation-based | Bayesian | |
|---|---|---|---|---|
| Fixed vs. random | \(\theta\) fixed, data random | data fixed, \(\theta\) ranked | resample/relabel data | data fixed, \(\theta\) random |
| Core object | sampling distribution of \(\hat\theta\) | likelihood \(L(\theta)\) | bootstrap / permutation distribution | posterior \(\pi(\theta \mid x)\) |
| Point summary | \(\hat p = 0.65\) | MLE \(= 0.65\) | resampled estimate | posterior mean \(\approx 0.636\) |
| Interval | 95% CI \((0.502, 0.798)\) | likelihood interval | bootstrap CI \(\approx (0.50, 0.80)\) | 95% credible \((0.493, 0.766)\) |
| Test / evidence | one-sided \(p \approx 0.029\) vs \(H_0:\theta=0.5\) | likelihood ratio \(0.65\) vs \(0.5\) | permutation \(p\) | \(P(\theta>0.5 \mid x) \approx 0.975\) |
| The claim | “95% of such intervals cover \(\theta\)” | “\(0.65\) best explains the data” | same, with fewer assumptions | “95% posterior probability \(\theta \in\) interval” |
| Key assumption | sampling model / normal approx. | a model for the data | exchangeability / resampling validity | a justified prior |
Key formulas
Standard error. For a proportion, \(\operatorname{SE}(\hat p) = \sqrt{\hat p(1-\hat p)/n}\); for a mean, \(\operatorname{SE}(\bar X) = s/\sqrt n\).
Confidence interval. \(\text{estimate} \pm (\text{critical value}) \times \operatorname{SE}\), with \(z^{*} = 1.96\) (proportion, 95%) or \(t_{n-1,\,0.975}\) (mean).
Test statistic and p-value. \(z = (\hat p - p_0)/\sqrt{p_0(1-p_0)/n}\); the p-value is the tail probability of \(z\) under \(H_0\).
Bias–variance. \(\operatorname{MSE} = \operatorname{Var} + \operatorname{Bias}^2\).
Bayes’ rule. \(\pi(\theta \mid x) \propto p(x \mid \theta)\,\pi(\theta)\); for a \(\text{Beta}(a,b)\) prior and \(x\) successes in \(n\) trials, the posterior is \(\text{Beta}(a+x,\ b+n-x)\) with mean \((a+x)/(a+b+n)\).
The recurring numbers (synthetic)
For the pass rate \(\theta\) (\(x = 26\), \(n = 40\)): \(\hat p = 0.65\); \(\operatorname{SE} \approx 0.075\); 95% CI \((0.502, 0.798)\); MLE \(0.65\); one-sided \(p \approx 0.029\); prior \(\text{Beta}(2,2) \to\) posterior \(\text{Beta}(28,16)\); posterior mean \(0.636\); 95% credible interval \((0.493, 0.766)\); \(P(\theta>0.5 \mid x) \approx 0.975\). For the mean gain (\(n = 36\), \(\bar x = 8.0\), \(s = 6.0\)): \(\operatorname{SE} = 1.0\); \(t\)-interval \((5.97, 10.03)\); bootstrap interval \(\approx (6.0, 10.0)\). For the two-group effect: difference \(d = 3.0\); permutation \(p \approx 0.04\).
The one thing to remember
When the frameworks print nearly the same numbers — and on an easy problem they often do — they still make different claims. A confidence interval is coverage of a procedure; a credible interval is a probability about \(\theta\). A p-value is a tail probability of the data under \(H_0\); a posterior tail probability is about \(\theta\) given the data. Read the claim, not just the digits.
This page is a study reference. For graded specifics — deadlines, submissions, and policies — Blackboard (the LMS) is authoritative. All numbers are synthetic and verified: false; the math gate is blocked pending sign-off.