 # 2. Model Selection: Bayesian Evidence for Dark Energy 30 points I...

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2. Model Selection: Bayesian Evidence for Dark Energy 30 points In Bayesian model selection, we compute the posterior odds between two models. Assuming no prior reason to favor one model over the other, the odds reduce to the Bayes Factor (MMID) p(D|Mi) p(Mi) = = p(Mj|D) p(D)M() p(Mi) 1 Given the free parameters 0 = {01,82, } of model Mi, we can write the likelihood of the model as (D)Mi) = (a) (10 points) Assuming OM + Dr = 1, compute p(D)M1) with free parameters DM, H0, and M. Use flat priors on the parameters such that OM € [0,3], H0 € [50, 100], and M € [18,20]. (b) (10 points) Fix OM = 1 and compute p(D)M2) with free parameters H0 and M with flat priors H0 € [50, 100], and M € [18,20]. (c) (10 points) Compute the Bayes Factor B12. Which model is preferred? How does this compare to the frequentist result computed using Wilks' Theorem?

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function de2()
global z m merr

z = T.Var1;
m = T.Var2;
merr = T.Var3;

p1 = pDM1()
p2 = pDM2()

return;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function p = pDM2()
global chi2m1
xm1 = OptModel2()
chi2m1 = chi2Model2(xm1)

lb = [50 -20];
ub = [100 -18];
lb = xm1 - [20 0.8];
ub = xm1 + [20 0.8];

Np = 20;
x0 = (-0.5 +[1:Np])/Np;
x1 = lb(1) + (ub(1)-lb(1))*x0;
x2 = lb(2) + (ub(2)-lb(2))*x0;
chi2 = zeros(Np,Np);
for i1 = 1:Np...

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