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Which is the correct equation for the posterior distribution?
(6.9) z + a N + a + b ︸ posterior = z N ︸ data N N + a + b ︸ weight + a a + b ︸ prior a + b N + a + b ︸ weight. Equation 6.9 indicates that the posterior mean is always somewhere between the prior mean and the proportion in the data.
How to calculate posterior distribution using Bayes rule?
One can use Gibbs and H-M models with help of JAGS and BUGS (Kruschke, 2014 ). Determining the posterior distribution directly from Bayes’ rule involves computing the evidence (a.k.a. marginal likelihood) in Equations 5.8 and 5.9. In the usual case of continuous parameters, the integral in Equation 5.9 can be impossible to solve analytically.
How is the simulation of posterior distribution difficult?
Simulation of posterior distribution values (sampling) directly is often difficult and challenging. Usually, most of the problems have intractable marginal posteriors and huge dimensionality that lead to different challenges in obtaining directly simulated values from the posteriors.
How are Mahalanobis distances plotted in a Q-Q plot?
A Q-Q plot can be used to picture the Mahalanobis distances for the sample. The basic idea is the same as for a normal probability plot. For multivariate data, we plot the ordered Mahalanobis distances versus estimated quantiles (percentiles) for a sample of size n from a chi-squared distribution with p degrees of freedom.
When does the prior mean influence the posterior mean?
Thus, the more data we have, the less is the influence of the prior, and the posterior mean gets closer to the proportion in the data. In particular, when N = a + b, the mixing weights are 0.5, which indicates that the prior mean and the data proportion have equal influence in the posterior.
When to use an ad hoc posterior distribution?
We may for example have an ad hoc estimate of the region of the parameter space where the true parameter value lies with 95% certainty. Then we just have to find a prior distribution whose 95% credible interval agrees with this estimate. But usually credible intervals are examined after observing the data.
Is the posterior mean symmetric about its mode?
• We have seen that the posterior distribution is • This normal distribution is symmetric about its mode, and the mean exists, the posterior mode and mean agree and equal • This is a weight average of the prior mean and the sample mean and lies between these two values.