The annual number of serious accidents in the mines of Great Britain were recorded every year from 1851 to 1962. We want to build a statistical model to evaluate if during those 112 years there has been a change in the rate of occurrence of events, which could perhaps be due to legislative changes to protect security.

Let (Y1, . . . , Ym1, Ym, Ym+1, . . . , Yn) be the number of annual incidents,

with Yi P oi() i = 1, 2, . . . , m Yj P oi() j = m + 1, . . . , n

The unknown parameters of the model are (, , m), while n is known and equal to 112.

Since the observations are supposed to follow a Poisson distribution, it is reasonable to consider Gamma priors for (, ): Gamma(, )

Gamma(a, b)

On the other side, the parameter m can have a uniform distribution: m Unif(0, n 1)

The data are the following: 1 y=c(4,5,4,1,0,4,3,4,0,6, 3,3,4,0,2,6,3,3,5,4,5,3,1,4,4,1,5,5,3,4,2,5,2,2,3,4,2,1,3,2, 1,1,1,1,1,3,0,0,1,0,1,1,0,0,3,1,0,3,2,2,0,1,1,1,0,1,0,1,0,0, 0,2,1,0,0,0,1,1,0,2,2,3,1,1,2,1,1,1,1,2,4,2,0,0,0,1,4,0,0,0, 1,0,0,0,0,0,1,0,0,1,0,0)

use MCMC algorithm to perform a Bayesian analysis for this model. It could be useful to procede with the following steps

write the likelihood function;

write the full posterior distribution;

code a Gibbs sampler with N = 10000 simulations and with a burnin of 1000 values;

plot the simulates marginal chain for each parameter and comment;

analyse the convergence of the chains with at least two tools;