In a Bayesian setting, the posterior distribution can be used as a prior distribution for a succeeding Bayesian λ)-x(M,...
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...
Bayesian Estimation and Confidence Regions: Concept Questions Σχ.-12 Consider the scenario where we observe n 10 random variable with sum AMAP M. Bayes Compare the values of the Bayes' estimator A0.833 and the maximum a-poster estimator -0.75 calculated earlier. Which of the following statements correctly describes the relative values and provides an accurate justification for it? (a) because this inequality will always hold regardless of the prior or model choice 1 ABaves AMAP (b) A< 1 because this inequality will...
ABayes Is the confidence Region R(a,b) symmetric around the Bayesian estimatorA , as defined earlier? Why or Why not? (a)Yes, because by construction, confidence intervals and hence confidence regions are symmetric around any consistent of the parameter (b)Yes, because our posterior distribution is symmetric and we chose a and b such that (-o0, a) and (b,oo) have an equal 5% Probability. (c)No, because our posterior distribution is not symmetric (it is either skewed to the left or skewed to the...
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...
Problem 4 - Bayesian inference with uniform prior The data are 21:n, the model is Normal(μ, σ*), with σ2 known. The problem is to obtain the posterior distribution of μ, p(p xỉ n, σ*)p(μ|xì n, σ2) when the prior po(A) is uniform in [-a, a] a. Using Bayes rule, obtain the expression of pĢi X1:n, σ*) as a function of a and the data. Be careful to handle all cases. Give and explicit simple expression for the normaliztion constant. You...
Consider the simplified Bayesian model for normal data The joint posterior pdf is ful, σ21 x)a(σ2,-/2-1 expl_jy.tx, _aPI The marginal posterior pdfs of μ and σ 2 can be obtained by integrating out the other variable (8.30) y@1 x) α (σ2)-m;,-1/2 expl-- Σ.-tri-x)2 (8.31) d. Let q1 and q2 be they/2 and 1-y/2 quantiles of (8.31). Show that the 1-γ credible interval (gi,q2) is identical to the classic confidence interval (5.19) (with ar replaced by y). Hence, a (1-α) stochastic...
in a Bayesian view. Consider the prior π(a)-1 for all a e R Consider a Gaussian linear model Y = aX+ E Determine whether each of the following statements is true or false. π(a) a uniform prior. (1) (a) True (b) False L(Y=y14=a,X=x) (2) π(a) is a jeffreys prior when we consider the likelihood (where we assume xis known) (a) True (b)False Y-XB+ σε where ε E R" is a random vector with Consider a linear regression model E[ε1-0, E[eErJ-1....
Problem 5 Let x ~ Pa, a Pareto distribution.... Problem 2. Consider 31, ... ,In ..d. N(0,0%) and the prior, T(0,0) = o-2(a+1)exp(-sa/20?). Compute and identify the posterior distribution 7(0,023,...,n) and show that it only depends on i and s? = 21-11; - 7)? Problem 3. Let ~ N(0,1). Assume that the prior median of @ is 0, the first quartile is -2, and the third quartile is +2. Then, if the prior distribution on 8 is of the form...
Discussion questions 1. What is the link between internal marketing and service quality in the airline industry? 2. What internal marketing programmes could British Airways put into place to avoid further internal unrest? What potential is there to extend auch programmes to external partners? 3. What challenges may BA face in implementing an internal marketing programme to deliver value to its customers? (1981)ǐn the context ofbank marketing ths theme has bon pururd by other, nashri oriented towards the identification of...