A Bayesian analysis produces a 95% credible interval of [1.2, 3.4] for a relative risk. What can you directly conclude?
AThere is a 95% chance the true relative risk is between 1.2 and 3.4, given the data and prior
BIf the study were repeated 100 times, 95 of the resulting intervals would contain the true value
CThe null hypothesis (RR = 1) is rejected at the 5% significance level
DThe p-value for the effect is less than 0.05
The 95% credible interval has the direct probabilistic interpretation: given the observed data and the prior, there is a 95% posterior probability that the parameter lies in this range. This is the statement practitioners actually want. Options B describes a frequentist confidence interval, which makes no probability statement about the fixed true parameter — it refers to the long-run behavior of the procedure. Options C and D conflate Bayesian credible intervals with null-hypothesis significance testing, which is a frequentist concept.
Question 2 Multiple Choice
A Bayesian epidemiologist runs an analysis with an informative prior derived from three prior studies. A colleague argues the results are invalid because of prior dependence. What is the most accurate response?
AThe prior makes the analysis invalid; only non-informative priors are scientifically acceptable
BThe prior is appropriate as long as it is substantively defensible and sensitivity analyses show robust conclusions across plausible priors
CInformative priors are only valid when the data are sparse; with sufficient data the prior is irrelevant regardless
DThe Bayesian analysis should be replaced with a frequentist meta-analysis to avoid subjectivity
Informative priors are scientifically legitimate when they reflect genuine prior knowledge from past studies or mechanistic reasoning. The appropriate response to concerns about prior influence is not to abandon the prior but to conduct sensitivity analyses — showing how conclusions change across a range of plausible priors. When results are robust (prior-robust), the evidence is convincing. When results depend heavily on the prior, this honestly reveals that data alone cannot resolve the question. Non-informative priors are not inherently more objective; they make their own implicit choices and can sometimes be poorly suited to constrained parameters like relative risks.
Question 3 True / False
When prior data are sparse and the observed dataset is small, the posterior distribution in a Bayesian analysis will be heavily influenced by the prior.
TTrue
FFalse
Answer: True
This is a core feature — and a core responsibility — of Bayesian inference. The posterior is proportional to the prior times the likelihood. When data are scarce (the likelihood is flat or weakly informative), the prior carries more weight in determining the posterior. When data are abundant, the likelihood dominates and the prior matters little. This is why prior specification is especially consequential in rare-disease epidemiology or small-sample studies, and why sensitivity analyses over different prior choices are essential in those settings.
Question 4 True / False
A Bayesian posterior probability directly answers questions like 'What is the probability the true effect exceeds a clinically meaningful threshold?' — something frequentist p-values can answer equally well.
TTrue
FFalse
Answer: False
This is a critical distinction. A frequentist p-value answers: 'What is the probability of observing data this extreme or more, assuming the null hypothesis is true?' It does not answer questions about the probability that the true parameter exceeds a threshold. A Bayesian posterior distribution does answer that question directly: you simply compute the posterior probability that the parameter exceeds the threshold of interest. This is why practitioners often find Bayesian outputs more naturally interpretable — they match the actual clinical or public health question being asked.
Question 5 Short Answer
In your own words, explain how a posterior distribution combines prior beliefs and observed data, and describe what it means for a Bayesian result to be 'prior-robust.'
Think about your answer, then reveal below.
Model answer: The posterior distribution is proportional to the prior distribution times the likelihood: P(θ|D) ∝ P(θ) × P(D|θ). The prior encodes beliefs about the parameter before observing the data; the likelihood reflects how probable the observed data are under each possible parameter value. The posterior combines these, weighting prior beliefs by how well they predict the observed data. A result is prior-robust when the posterior conclusions (e.g., the credible interval, or the posterior probability that the effect exceeds a threshold) remain substantively similar across a range of defensible prior specifications — informative, weakly informative, and non-informative. Prior robustness means the data are sufficiently informative to override prior differences, making the conclusions credible to analysts who hold different prior beliefs.
Prior robustness is the Bayesian analog of sensitivity analysis in frequentist work. It doesn't mean the prior is irrelevant — it means the data are strong enough that reasonable prior disagreements don't change the conclusions. When results are not robust across priors, this is itself a scientifically important finding: it means the study alone cannot resolve the question, and more data or clearer prior knowledge is needed.