Bayes vs. classical cheatsheet
What each framework claims, and what the claim means
A side-by-side reference for the comparison we develop carefully in Week 12. The goal is not to declare a winner — it is to say precisely what each kind of statement means and what it assumes. This is original course material.
The core difference in one line
- Classical (frequentist): the parameter \(\theta\) is a fixed unknown; probability describes the long-run behavior of procedures over hypothetical repeated samples.
- Bayesian: \(\theta\) is uncertain and described by a distribution; probability describes our plausibility for \(\theta\) given the data and the prior.
Side by side
| Question | Classical answer | Bayesian answer |
|---|---|---|
| What is \(\theta\)? | a fixed constant | a quantity with a posterior distribution \(f(\theta \mid y)\) |
| What is random? | the data / the procedure | our uncertainty about \(\theta\) (given the data) |
| Interval | a confidence interval: 95% of such intervals cover \(\theta\) in the long run | a credible interval: 95% posterior probability that \(\theta\) is in \([L, U]\) |
| “Evidence against \(H_0\)” | a p-value: \(P(\text{data this extreme} \mid H_0)\) | a posterior probability or Bayes factor about the hypotheses |
| Prediction | a point/interval from the fitted model | a posterior predictive distribution that averages over parameter uncertainty |
| Role of prior assumptions | implicit (model + procedure choices) | explicit (the prior \(f(\theta)\), stated and checkable) |
The interpretations students most often swap
- A confidence interval is not a 95% probability that \(\theta\) is inside it. That probability statement is the credible interval’s, and it requires a prior.
- A p-value is not \(P(H_0 \mid \text{data})\). It is \(P(\text{data} \mid H_0)\) — a statement about the data under a hypothesis, not about the hypothesis given the data.
- “The data prove \(H_1\)” overclaims in both frameworks. Data shift plausibility / provide evidence; they do not prove.
When the two nearly agree
With a lot of data and a weak prior, the posterior is dominated by the likelihood, and a Bayesian credible interval and a classical confidence interval often land in nearly the same place — for very different reasons. The numbers can match while the meaning of the interval differs. We return to this in Weeks 5 and 12.
Public vs. graded
A public reference page. Graded comparisons and their keys are authoritative in the LMS (Blackboard).