How it works...
The output that we produced here looks reasonably fine, because the posterior density seems to be oscillating slightly below 1 (this is caused because even though the coefficient is equal to 1, our priors have a mean of 0.5). It's worth noting that the proposal density (the function that proposed and generates random values) should be tuned in order to achieve a 25% acceptance rate (all Bayesian MCMC packages do this).
It’s also worth noting that there is no unanimous consensus regarding how thinning should be done, to the extent that some practitioners don’t even use it. They recommend running the chains for very long periods until the autocorrelation fades out, or even using the MCMC with autocorrelation.
An interesting ...
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