Highest Posterior Density Region and Central Credible Region

Question

Given a posterior p(Θ|D) over some parameters Θ, one can define the following:

Highest Posterior Density Region:

The Highest Posterior Density Region

Answer 1

PyMC has a built in function for computing the hpd. In v2.3 it's in utils. See the source here. As an example of a linear model and it's HPD

import pymc as pc  
import numpy as np
import matplotlib.pyplot as plt 
## data
np.random.seed(1)
x = np.array(range(0,50))
y = np.random.uniform(low=0.0, high=40.0, size=50)
y = 2*x+y
## plt.scatter(x,y)

## priors
emm = pc.Uniform('m', -100.0, 100.0, value=0)
cee = pc.Uniform('c', -100.0, 100.0, value=0) 

#linear-model
@pc.deterministic(plot=False)
def lin_mod(x=x, cee=cee, emm=emm):
    return emm*x + cee 

#likelihood
llhy = pc.Normal('y', mu=lin_mod, tau=1.0/(10.0**2), value=y, observed=True)

linearModel = pc.Model( [llhy, lin_mod, emm, cee] )
MCMClinear = pc.MCMC( linearModel)
MCMClinear.sample(10000,burn=5000,thin=5)
linear_output=MCMClinear.stats()

## pc.Matplot.plot(MCMClinear)
## print HPD using the trace of each parameter 
print(pc.utils.hpd(MCMClinear.trace('m')[:] , 1.- 0.95))
print(pc.utils.hpd(MCMClinear.trace('c')[:] , 1.- 0.95))

You may also consider calculating the quantiles

print(linear_output['m']['quantiles'])
print(linear_output['c']['quantiles'])

where I think if you just take the 2.5% to 97.5% values you get your 95% central credible interval.