In contrast to traditional REML methods, using regression of breeding values based on posterior distribution allows a conservative estimate of evolutionary trend (Hadfield et al. 2010).
Given a likelihood function, f(θ|D), where θ denotes parameter of interest and D denotes observed data, and prior distribution, p, on the parameter space, Θ, our statistical inference is based on posterior distribution of θ, p(θ|D)∝f(θ|D p.
Application of the Bayesian framework to change point estimation provides a way of making a set of inferences based on posterior distributions for the time and the magnitude of a change as well as assessing the validity of underlying assumptions in the change point model itself (Gelman et al. 2004).
However, it depends on prior input, and this dependence does not decrease as sample size increases, unlike parameter's estimation based on posterior distributions [ 20].
Gibbs sampling based on posterior distributions conditional on other effects was implemented for estimation by averaging the samples from 10,000 cycles, after discarding the first 1,000.
Parameter estimates were based on posterior probability distributions constructed by sampling the stationary distribution for 40,000,000 generations, sampling every 1000 steps.
In the Bayesian statistical framework, the inferences are based on posterior probability distributions of the quantities of interest (Gelman et al., 2004).
Furthermore, we have added a section in Supplement file 1 ("Initial conditions") detailing the initial conditions, and we have extended the section "In silico predictions" to describe techniques used for simulations based on posterior parameter distributions.
Sorensen & Waagepetersen [ 6] have applied a Markov chain Monte Carlo (MCMC) algorithm to estimate the parameters in a similar model, which has the advantage of producing model-checking tools based on posterior predictive distributions and model-selection criteria based on Bayes factor and deviances.
To be more robust, we avoid using a point estimator for directly, but use the expected value of based on the posterior distribution of (16) (17) where shape and scale parameterize the marginal posterior distribution of as in Equation 14.
In Bayesian analysis inferences are based on the posterior distribution of the unknowns given the data with the general form of the posterior density is given in (3).
