Following Bayes's rule, the posterior distribution
is [pi](p|Data)~D([alpha]+c), where c = ([c.
The common idea of methods which are developed to attack this problem is to construct an approximate posterior distribution
The joint posterior distribution
of the parameters [phi] given the data D is equal to
To answer the question about how strongly alpha diversity is related to habitat complexity, or to predict alpha diversity from habitat complexity, we drawn inferences from a Bayesian framework, where is key the joint posterior distribution
of [beta] (vector that contains [[beta].
3) He finds, using a Bayesian analysis, that the data are essentially completely uninformative regarding key parameters governing the evolution of the natural rate of interest: The posterior distribution
on the properties of the natural rate of interest process--for example, the variance of the innovation to the natural rate of interest--essentially equals the prior distribution.
On the other hand, simulation algorithms consist of taking samples of parameters from the posterior distribution
using a Markov chain Monte Carlo algorithm implemented via Gibbs sampling (Gilks, Richardson &
Combining the joint prior and the likelihood function using Bayes theorem we get the following posterior distribution
Obviously, the posterior distribution
of [theta] is Gamma distribution, i.
In this study, a prior being "uninformative" means that the posterior distribution
is proportional to the likelihood.
After performing an initial burn-in, we stored and summarized 1,000 values from the posterior distribution
of each parameter by using descriptive statistics (posterior mean, 95% posterior credible interval [95% CrI], and p value).
We used three chains to check for the stability of posterior distribution
estimates using Gelman and Rubin's convergence diagnostic (Gelman and Rubin 1992).
Subsequently, a posterior distribution
is estimated to carry out inference on the parameter values.