In Bayes' theorem, which statement about the posterior probability is true?

In Bayes' theorem, which statement about the posterior probability is true?

• A. Posterior equals prior.
• B. Posterior is proportional to prior times likelihood, normalized by the total probability of data.
• C. Posterior does not depend on data.
• D. Posterior is always larger than prior.

Answer: (B)

Bayes’ theorem updates what we believe about a parameter after seeing data. The posterior belief is obtained by taking the prior belief and weighting it by how likely the observed data are under each possible parameter value (the likelihood). Then you normalize by the total probability of the data to make sure the probabilities across all parameter values sum to one. In formula form, p(θ|data) = [p(data|θ) p(θ)] / p(data), where p(data) is the marginal likelihood (the integral or sum of p(data|θ) p(θ) over all θ). This normalization is what turns the proportional relationship into an actual probability distribution. This is why the statement in question is true: the posterior is proportional to the prior times the likelihood, scaled by the total probability of the data. The other options don’t fit because the posterior changes with data, is not simply equal to the prior, and need not be larger than the prior—the data can push beliefs up or down depending on how likely the observed data are under different parameter values.

Bayes’ theorem updates what we believe about a parameter after seeing data. The posterior belief is obtained by taking the prior belief and weighting it by how likely the observed data are under each possible parameter value (the likelihood). Then you normalize by the total probability of the data to make sure the probabilities across all parameter values sum to one. In formula form, p(θ|data) = [p(data|θ) p(θ)] / p(data), where p(data) is the marginal likelihood (the integral or sum of p(data|θ) p(θ) over all θ). This normalization is what turns the proportional relationship into an actual probability distribution.

This is why the statement in question is true: the posterior is proportional to the prior times the likelihood, scaled by the total probability of the data. The other options don’t fit because the posterior changes with data, is not simply equal to the prior, and need not be larger than the prior—the data can push beliefs up or down depending on how likely the observed data are under different parameter values.