21.7 Problems

a. Assume the probability density fx(·|θ) is determined by a parameter vector ψ which is depending on the regression parameters θ = (θ0, θ1,…,θk), i.e. fxi(·|θ) = g(·|ψ(θ, xi)).

Which form has the likelihood function for observed data (x1, y1),…,(xn, yn)? How does Bayes’ theorem look in this situation?

b. Let the dependent quantity Yx have an exponential distribution Exτ with density Assuming a regression line θ0 + θ1 · x for the mean value τ, i.e. τ(x) = θ0 + θ1 · x and therefore calculate the likelihood function and write down Bayes’ theorem for this special case for data points (x1, y1),…,(xn, yn) in the plane 2.

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