Final answer:
The prior distribution for a parameter θ conveys existing knowledge before data collection and is combined with data to form the posterior distribution. For a binomial distribution with a beta prior, the posterior is also beta distributed and the set of beta distributions is called a conjugate family. The Bayes estimator under squared error loss can be computed as the mean of the posterior distribution.
Step-by-step explanation:
The purpose of a prior distribution for a parameter θ is to quantify existing knowledge about the parameter before data is collected. This knowledge can be based on previous studies or expert opinion and is used to inform the Bayesian analysis. The prior information can help provide higher certainty in parameter estimates.
The posterior distribution of θ given x in the case of a binomial distribution with a beta prior is also a beta distribution. The process of determining this involves using Bayes' theorem, where the likelihood of the observed data is combined with the prior distribution to produce the posterior distribution. If X is a binomial random variable X ~ Bin(n, θ) and the prior distribution for θ is Beta(a, β), then the posterior distribution for θ after observing x successes in n trials is Beta(a+x, β+n-x).
A conjugate family for the binomial distribution is the set of beta distributions, because the posterior distribution is in the same family as the prior distribution, which simplifies the process of Bayesian updating.
If the prior for θ is Beta(1, 1), which is the uniform distribution U(0, 1), and the likelihood is based on a binomial distribution, the Bayes estimator for θ under the squared error loss is simply the mean of the posterior distribution, which is μ = (a+x)/(a+β+n).