D Sporsmal 1 1 poeng Which sources of information are used in classical (frequentist) statistics? O Data and prior information on parameters Something else O Prior information on parameters only Data only D Sporsmal 2 1 poeng Which sources of information are used in Bayesian statistics? O Data only O Prior information on parameters only Something else Data and prior information on parameters D Sporsmal 3 1 poeng Which distribution does the density kernel h(x) = exp (4 - zx2) correspond to? O N(1, 1) O N(0, 22 ) O X3 O Poisson Distribution with mean 8Sporsmal 4 1 poeng What is the conjugate prior for / when data are distributed as N(, o2) (with a known)? O Normal distribution O Uniform distribution O Weibull distribution O Gamma distribution D Sporsmal 5 1 poeng Suppose you have observations with and exponential distribution with mean 1/ 1. What is the conjugate prior for X? O Normal distribution O Gamma distribution Poisson distribution O Log-normal distribution D Sporsmal 6 1 poeng Consider the situation in exercise 2 from the note on Bayesian statistics. Re-do the exercise with prior p(0) = rexp(-Or) for a constantr > 0. What is the posterior distribution you obtain then? O Something else Gamma distribution with parameters n + r - land (r + Cl_, t;)-1 Gamma distribution with parameters n + 1 and (r + >!_, t;) 1 Gamma distribution with parameters n + r - 1 and (2 + >7_, t;)Note on Bayesian statistics, exercise 2 With the exponential density written as f(t|0) = de- we get the likelihood n n L(0) = II f(t:10) = II de otz = one Enti i=1 i=1 The prior density is given as p(0) = 2e-2. We then get the following posterior distribu- tion: p(0|data) = c . L(0) . p(0) = cone-Zizitize-20 = contl-1e-0(Li=tit2) We see that this (as a function of 0) is on the same form as a gamma distribution with parameters n + 1 and 1/()_t; + 2). Le. the distribution is a gamma distribution with parameters n + 1 and 1/(27 , ti + 2)