On this page, you will be given a distribution and another distribution conditional on the first one. Then, you will find the posterior distribution in a Bayesian approach. You will compute the Bayesian estimator, which is defined in lecture as the mean of the posterior distribution. Then, determine if the Bayesian estimator is consistent and/or asymptotically normal.
We recall that the Gamma distribution with parameters \, q>0 \, and \, \lambda >0 \, is the continuous distribution on \, (0,\infty ) \, whose density is given by \, \displaystyle f(x)=\frac{\lambda ^ q x^{q-1}e^{-\lambda x}}{\Gamma (q)} \,, where \, \Gamma \, is the Euler Gamma function \, \Gamma (q)=\int _0^\infty t^{q-1} e^{-t} dt \,, and its mean is \, q/\lambda \,.
We also recall that the \, \textsf{Beta}(a,b) \, distribution has the density \, \displaystyle f(x) = \frac{x^{a-1}(1-x)^{b-1}}{B(a,b)} \, and expectation \, a/(a+b) \,, where \, \displaystyle B(a,b) = \frac{\Gamma (a) \Gamma (b)}{\Gamma (a+b)} \,.
\, p\sim \textsf{Beta}(a,b) \, for some \, a,b>0 \, and conditional on \, p \,, \, X_1,\ldots ,X_ n\stackrel{i.i.d.}{\sim } \textsf{Ber}(p) \,.
What is the Bayesian estimator \hat{p}^{\text {Bayes}}?
(If applicable, enter barX_n for \bar{X_ n},\, max_n for \displaystyle \text {max}_{i=1\ldots n} X_ i. )
\hat{p}^{\text {Bayes}}=\quad
unanswered
{::}
Determine whether the Bayesian estimator is consistent, and whether it is asymptotically normal.
Consistent and asymptotically normal
Consistent but not asymptotically normal
Asymptotically normal but not consistent
Neither consistent nor asymptotically normal
unanswered
If it is asymptotically normal, what is its asymptotic variance V(a,b,p)? If it is not asymptotically normal, type in \, 0 \,.
V(a,b,p)=\quad
To find the Bayesian estimator, we need to find the posterior distribution and then calculate its mean.
Given that \, p \, follows a \, \textsf{Beta}(a,b) \, distribution and \, X_1,\ldots ,X_ n\stackrel{i.i.d.}{\sim } \textsf{Ber}(p), the likelihood function is given by:
\, \displaystyle L(p;X_1,\ldots ,X_ n) = \prod _{i=1}^ n p^{X_ i}(1-p)^{1-X_ i}
The prior distribution is given by \, \textsf{Beta}(a,b).
The posterior distribution is given by:
\, \displaystyle p|X_1,\ldots ,X_ n \sim \textsf{Beta}(a+\sum _{i=1}^ n X_ i, b+n-\sum _{i=1}^ n X_ i)
The mean of the posterior distribution (Bayesian estimator) is:
\, \displaystyle \hat{p}^{\text {Bayes}} = \frac{a+\sum _{i=1}^ n X_ i}{a+b+n}
Now we need to determine if the Bayesian estimator is consistent and/or asymptotically normal.
To show consistency, we need to show that the Bayesian estimator converges to the true value of \, p \, as \, n \, increases.
To show asymptotic normality, we need to show that the Bayesian estimator follows a normal distribution as \, n \, increases, with mean equal to the true value of \, p \, and variance equal to some finite value.
Unfortunately, we do not have enough information on the values of \, a \, and \, b \, to determine if the Bayesian estimator is consistent and/or asymptotically normal. Therefore, we cannot compute the asymptotic variance. The answer is "unanswered".
To find the Bayesian estimator, we need to calculate the posterior distribution first.
Given that p follows a Beta(a, b) distribution, and X₁, X₂, ..., Xₙ are independent and identically distributed random variables following a Bernoulli distribution with parameter p, the likelihood function is:
L(p|x₁, x₂, ..., xₙ) = p^∑xᵢ(1-p)^(n-∑xᵢ)
The prior distribution on p is a Beta(a, b) distribution, so the prior density function is:
f(p) = p^(a-1)(1-p)^(b-1)/B(a, b)
Using Bayes' theorem, the posterior distribution is proportional to the product of the likelihood and the prior:
f(p|x₁, x₂, ..., xₙ) ∝ L(p|x₁, x₂, ..., xₙ)f(p)
∝ p^∑xᵢ(1-p)^(n-∑xᵢ) * p^(a-1)(1-p)^(b-1)
∝ p^(∑xᵢ + a - 1)(1-p)^(n - ∑xᵢ + b - 1)
The above expression is proportional to a Beta distribution with parameters (α, β), where α = ∑xᵢ + a and β = n - ∑xᵢ + b.
The Bayesian estimator, denoted by ̂p^Bayes, is equal to the mean of the posterior distribution.
Thus, ̂p^Bayes = (α/(α+β)) = (∑xᵢ + a) / (n + a + b).
To determine if the Bayesian estimator is consistent and/or asymptotically normal:
Consistency: A Bayesian estimator is consistent if it converges to the true parameter value as the sample size increases. In this case, the true parameter value is p. If ∑xᵢ/n converges to p, then the Bayesian estimator is consistent.
Asymptotic Normality: A Bayesian estimator is asymptotically normal if, as the sample size approaches infinity, the distribution of the estimator approaches a normal distribution centered around the true parameter value with some variance.
To find the asymptotic variance, V(a, b, p), we need more information about the prior distribution.
Please provide the values of a and b for the Beta(a, b) prior distribution.