Q15. Prior beliefs and conflicting data
[3 marks]
A researcher models independent observations as Bernoulli with unknown success probability \theta. The researcher chooses a prior that is uniform on (0,1/2).
Of 100 observed trials, 80 are successes. The researcher argues that these data should make the posterior probability of \theta>1/2 large.
Is this reasoning correct? Explain what happens and whether collecting more observations can overcome the restriction imposed by this prior.
Definitions. Let L(\theta;{\boldsymbol{x}})=p({\boldsymbol{x}}\mid\theta) and let m({\boldsymbol{x}})=\int L(\theta;{\boldsymbol{x}})p(\theta)\,d\theta be the marginal likelihood. Bayes’ update is p(\theta\mid{\boldsymbol{x}})=L(\theta;{\boldsymbol{x}})p(\theta)/m({\boldsymbol{x}}), provided 0<m({\boldsymbol{x}})<\infty. The prior here excludes the entire region \theta>1/2, not merely a single value.
The reasoning is incorrect. The prior density is zero for every \theta>1/2. Bayes’ theorem gives p(\theta\mid{\boldsymbol{x}}) \propto p({\boldsymbol{x}}\mid\theta)p(\theta), so p(\theta\mid{\boldsymbol{x}})=0 \qquad\text{whenever }\theta>1/2. Hence \boxed{ \mathbb{P}(\theta>1/2\mid{\boldsymbol{x}})=0. } This remains true regardless of how many observations are collected. The likelihood can redistribute posterior mass only among parameter values to which the prior assigns positive density; it cannot create support in a region excluded by the prior.
The observed proportion 80/100=0.8 instead reveals serious conflict between the data and the assumed model–prior combination. The researcher should reconsider the restricted prior, or investigate whether the Bernoulli model is appropriate. With the restriction left unchanged, additional data cannot make the posterior probability above 1/2 positive.