Let ?1, ... , ?10 are identically independently distributed (iid) with Gamma(2, ?)
a) Compute the likelihood function (LF).
b) Adopt the appropriate conjugate prior to the parameter ? (Hint:
Choose hyperparameters optionally within the support of
distribution).
c) Using (a) and (b), find the posterior distribution of ?. d) Compute the minimum Bayesian risk estimator of ?.
Let ?1, ... , ?10 are identically independently distributed (iid) with Gamma(2, ?) a) Compute the...
Let ?1, … , ?10 are identically independently distributed (iid) with an Exponential model with parameter ( λ ), a) Compute the likelihood function (LF). b) Adopt the appropriate conjugate prior to the parameter λ (Hint: Choose hyperparameters optionally within the support of distribution). c) Using (a) and (b), find the posterior distribution of λ. d) Compute the minimum Bayesian risk estimator of λ.
Let X1, . . . , Xn be a random sample following Gamma(2, β) for some unknown parameter β > 0. (i) Now let’s think like a Bayesian. Consider a prior distribution of β ∼ Gamma(a, b) for some a, b > 0. Derive the posterior distribution of β given (X1, . . . , Xn) = (x1,...,xn). (j) What is the posterior Bayes estimator of β assuming squared error loss?
Let X1,…Xn ~ iid Gamma (α, θ) where the α is known and interested in the rate parameter θ, and we chosen a prior θ~ Gamma (3, 1). Find the posterior distribution
Question 2 a. Show that, for the exponential model with gamma prior, the posterior Π(9121m) under n observations can be computed as the posterior given a single observation xn using the prior q(の는 1101r1:n-1). Give the formula for the parameters (an,ßn) of the posterior ll(θ|X1:n, α0,Ao) as a function of (an-1, Bn-1). b. Visualize the gradual change of shape of the posterior II(01:n, ao, Bo) with increasing n: . Generate n 256 exponentially distributed samples with parameter θ-1. . Use...
Let Xi, , xn independent identically Gumbel!(-10g(A), 1) distributed. Let parameter λ (0.00) be unknown. (a) Show: λ,--n 1 (b) Explain: Is λη unbiased? (c) Explain: Is An consistent? e_Λ ls the maximum likelihood estimator for λ .
One side concept introduced introduced in the second Bayesian lecture is the conjugate prior. Simply put, a prior distribution π (0) is called conjugate to the data model, given by the likelihoodfunction L (Xi θ if the posterior distribution π (ex 2, , . , X ) is part of the same distribution family as the prior. This problem will give you some more practice on computing posterior distributions, where we make use of the proportionality notation. It would be...
2. Asymptotic Maximum Likelihood. 25 Let X1, ..., Xn be independently Poisson distributed with parameter 1, i.e. fx, (x) = exto is X= 0, 1, 2, ... =0,1,2,... (a) Derive the maximum likelihood estimator în of 1 based on n measurements. 5 (b) Show that în is consistent. 5 (c) Is în (asymptotically) efficient? 5 (d) Derive the asymptotic distribution of vn(în – 1). 10
Suppose observations X1, X2,.. are recorded. We assume these to be conditionally independent and exponen- tially distributed given a parameter θ: Xi ~' Exponential(θ), for all i 1, . . . , n. The exponential distribution is controlled by one rate parameter θ > 0, and its density is for r ER+ 1. Plot the graph of p(x:0) for θ 1 in the interval x E [0,4] 2. What is the visual representation of the likelihood of individual data points?...
Recall that the exponential distribution with parameter A > 0 has density g (x) Ae, (x > 0). We write X Exp (A) when a random variable X has this distribution. The Gamma distribution with positive parameters a (shape), B (rate) has density h (x) ox r e , (r > 0). and has expectation.We write X~ Gamma (a, B) when a random variable X has this distribution Suppose we have independent and identically distributed random variables X1,..., Xn, that...
Exercise 7 (Ancilliarity) Choose one: 1. Let {X;} –1 be independent and identically distributed observations from a location paramter family with cumulative distribution function F(x – 0), -00 < 0 < 0. Show that range of the distribution of R = maxi(Xi) – mini(Xi) does not depend on the parameter 8.) Hint: Use the facts that X1 = Z1 + 0 , ..., Xin = Zn + 0 and mini(Xi) = mini(Zi + 0), maxi(Xi) = maxi(Z; +0), where {Zi}=1...