How to calculate posterior predictive distribution?
For a new data point x∗, we can compute P(x∗) using:
- a fully bayesian approach: the posterior predictive distribution P(x∗|X)=∫P(θ|X)P(x∗|θ)dθ
- the likelihood parameterized by the maximum a posteriori estimate: P(x∗|θMAP), where θMAP=argmaxθP(θ|X)
What is a prior predictive distribution?
The prior predictive distribution, pa,b(x), is the beta-binomial distribution with parameters (n,a,b). This discrete distribution gives the probability of getting k successes out of n trials given the hyper-parameters (a,b) on the probability of success.
How is the posterior predictive distribution used in Bayesian statistics?
In Bayesian statistics, the posterior predictive distribution is the distribution of possible unobserved values conditional on the observed values. Given a set of N i.i.d. observations, a new value will be drawn from a distribution that depends on a parameter
Is the posterior predictive distribution the same as the compound distribution?
This shows that the posterior predictive distribution of a series of observations, in the case where the observations follow an exponential family with the appropriate conjugate prior, has the same probability density as the compound distribution, with parameters as specified above. The observations themselves enter only in the form
Why is the integral of a posterior distribution tractable?
The reason the integral is tractable is that it involves computing the normalization constant of a density defined by the product of a prior distribution and a likelihood. When the two are conjugate, the product is a posterior distribution, and by assumption, the normalization constant of this distribution is known.
When is a distribution of posteriors Bayes plausible?
A distribution of posteriors is Bayes-plausible if the expected posterior probability of each state equals its prior probability: ∫ μ d τ ( μ) = μ 0.