Prior p(θ)Likelihood p(X|θ)Posterior p(θ|X)When we are employing conjugate distributions, (at least) which of the above three densities should have the same form?Group of answer choicesPrior and likelihoodPrior and posteriorLikelihood and posterior

Which one of the terms below is a joint probability?Group of answer choicesP(θ)P(D)P(θ,D)P(θ|D)

posterior probability

Suppose you have data x coming from a N (µ, σ2) distribution where the variance σ2 isknown but you want to perform Bayesian inference on the mean µ. We are going toconsider two different scenarios of prior knowledge:• The first scenario involves using a normal prior i.e. π(µ) = N (ν, w) and lettingw → ∞ to represent lack of knowledge.• In the second scenario, you know that there is probability p that µ = 0, but there islittle prior knowledge about µ̸ = 0. This prior information will be represented by amixture distribution, with a discrete probability at µ = 0 (i.e. a probability atom)and π(µ) = N (0, w) for µ̸ = 0 and let w → ∞ to represent our lack of knowledge forµ̸ = 0.(a) [6 marks] Analytically calculate the posterior probability π(µ|x) for the first sce-nario. Comment on whether the prior and posterior are proper or improper.(b) [5 marks] Analytically calculate the posterior probability π(µ = 0|x) for the sec-ond scenario. Give an explanation in words (1-2 sentences) of what this posteriorindicates and whether this is sensible.(c) [3 marks] Give an explanation as to what properties of the prior in (b) give rise tothe behaviour of the posterior in (b), compared to the prior in (a).(d) [2 marks] Suggest an alternative prior in the spirit of (b) and show that it gives amore sensible posterior π(µ = 0|x) than the one calculated in (b)

Pour a et b strictement positifs fix´es, on consid`ere le cadre bay´esien suivantθ ∼ Γ(a, b) = πX = (X1, ..., Xn) | θ ∼ P(θ)⊗n1. D´eterminer la loi a posteriori π[ | X]. Commenter.

Posterior means _________________ and is opposite to ____________________.Multiple choice question.situated behind; anteriorventral; supinesituated away from the center; proximalsituated below; superiornearer the head; caudal