Prior
Definition: A probability distribution encoding parameter support and assumptions before conditioning on current data.
Why it matters here: It helps answer How do data, prior assumptions, and a finite Monte Carlo sample jointly shape a posterior? within the numerical model described here.
Relationships: Connect it to Posterior, the governing expression \(p(\theta\mid D)\propto p(D\mid\theta)p(\theta),\quad p\mid k,n\sim\mathrm{Beta}(k+\alpha,n-k+\beta)\), and the computed diagnostic developed later in the lecture.
Boundary or common confusion: A narrow posterior can still be wrong if the likelihood omits dependence, selection, calibration, or model discrepancy. Prior sensitivity should be tested when data are weak.
Posterior
Definition: The parameter distribution obtained by combining likelihood and prior through Bayes' rule.
Why it matters here: It helps answer How do data, prior assumptions, and a finite Monte Carlo sample jointly shape a posterior? within the numerical model described here.
Relationships: Connect it to Conjugate model, the governing expression \(p(\theta\mid D)\propto p(D\mid\theta)p(\theta),\quad p\mid k,n\sim\mathrm{Beta}(k+\alpha,n-k+\beta)\), and the computed diagnostic developed later in the lecture.
Boundary or common confusion: A narrow posterior can still be wrong if the likelihood omits dependence, selection, calibration, or model discrepancy. Prior sensitivity should be tested when data are weak.
Conjugate model
Definition: A likelihood-prior pairing whose posterior remains in the same distribution family.
Why it matters here: It helps answer How do data, prior assumptions, and a finite Monte Carlo sample jointly shape a posterior? within the numerical model described here.
Relationships: Connect it to Credible interval, the governing expression \(p(\theta\mid D)\propto p(D\mid\theta)p(\theta),\quad p\mid k,n\sim\mathrm{Beta}(k+\alpha,n-k+\beta)\), and the computed diagnostic developed later in the lecture.
Boundary or common confusion: A narrow posterior can still be wrong if the likelihood omits dependence, selection, calibration, or model discrepancy. Prior sensitivity should be tested when data are weak.
Credible interval
Definition: A posterior probability interval conditional on the adopted model and prior.
Why it matters here: It helps answer How do data, prior assumptions, and a finite Monte Carlo sample jointly shape a posterior? within the numerical model described here.
Relationships: Connect it to Posterior predictive check, the governing expression \(p(\theta\mid D)\propto p(D\mid\theta)p(\theta),\quad p\mid k,n\sim\mathrm{Beta}(k+\alpha,n-k+\beta)\), and the computed diagnostic developed later in the lecture.
Boundary or common confusion: A narrow posterior can still be wrong if the likelihood omits dependence, selection, calibration, or model discrepancy. Prior sensitivity should be tested when data are weak.
Posterior predictive check
Definition: A comparison of replicated data drawn from the fitted model with observed diagnostic structure.
Why it matters here: It helps answer How do data, prior assumptions, and a finite Monte Carlo sample jointly shape a posterior? within the numerical model described here.
Relationships: Connect it to Prior, the governing expression \(p(\theta\mid D)\propto p(D\mid\theta)p(\theta),\quad p\mid k,n\sim\mathrm{Beta}(k+\alpha,n-k+\beta)\), and the computed diagnostic developed later in the lecture.
Boundary or common confusion: A narrow posterior can still be wrong if the likelihood omits dependence, selection, calibration, or model discrepancy. Prior sensitivity should be tested when data are weak.