Bonjour ! Je suis actuellement en L3 en Ecosse et nous travaillons sur l'inférence Bayésienne. En ce moment nous avons un projet à réalisé mais j'ai du mal à arriver au bout des questions et à vraiment comprendre ce que je fait :/

Je vais recopier le sujet (qui est en anglais) :

You are a Bayesian trainee actuary working for an insurance company and have been asked to estimate how much capital the company should hold in reserve to cover possible claims arising from a certain portfolio of policies over the next 12 months. You are asked to specify the amount for which the company can be 95% certain that the aggregated claim amount will not exceed this value. You are advised to assume that the aggregated claim amount X ∈ R+ in any given year follows a Rayleigh distribution, with probability density function given by
p(x|θ) = x θ exp{−θx2/2} , x > 0, with unknown parameter θ ∈ R+.
Given θ, the claim amounts in different years are assumed independent of each other.
The portfolio represents a new business area for the company and the only data you have is the total claim size per year from the previous n = 4 years, during which there were aggregated claim amounts of x1 = £1.1M, x2 = £0.7M, x3 = £0.45M, and x4 = £1.3M.
Given your lack of prior experience with policies of this kind you decide to use a suitably
chosen non-informative or objective prior distribution for θ.

1. Identify a suitable prior π(θ) and briefly justify your choice.
2. Derive the posterior density π(θ|x), presenting appropriate graphical or other summaries of the distribution.
3. Now let Y denote the aggregated claim amount in the next 12 months. Derive the posterior predictive distribution π(y|x) (you may express predictive distributions as infinite sums and integrals).
4. Use simulation in R to generate a suitably large random sample from π(y|x). Report
appropriate graphical and quantitative summaries of the distribution.
Hint: Given a sample (a1, b1), . . . ,(an, bn) from a distribution p(a, b), the sample associated with the first component, (a1, . . . , an), is distributed according to the marginal distribution p(a) = integral p(a, b)db. Moreover, you may draw samples from p(a, b) by first
drawing samples from p(b) and subsequently from p(a|b).

Alors pour la question 1 j'ai utilisé un Jeffrey's prior qui correspond à la racine carrée de l'information de Fisher pour arriver au prior g(θ) = 1/ θ

Ensuite pour la question 2 j'ai la vraissemblance (likelihood) suivante :
L(x; θ) = xi θⁿ exp{(-θ xi²)/2}
Pour arriver au posterior suivant : π(θ|x) = xi θ^n-1 exp{(-θ xi²)/2}

A voir si tout ça est correct :/

Ensuite c'est pour la question 3 que je bloque. Je ne sais pas du tout quel calcul poser :/ Si quelqu'un pouvait me donner une piste.
A et aussi pour la question 4, simulation sur R, si quelqu'un si connait en simulation Monte Carlo je lui en serait très reconnaissante de mon donner quelques indices astuces

Merci beaucoup à ces qui prendront le temps et la peine de m'aider