def generate_pi_samples_mh(alpha,n_samples,data,beta):
# starttime = time.time()
K = data.shape[0]
# randomly initialize pi
pi = np.random.dirichlet(alpha * np.ones(K))
current_val = log_censored_dirichlet_density(pi,alpha=alpha,data=data)
samples = []
n_accepts = 0
# loop mh proposals
n_total = 0
while len(samples) < n_samples:
### make a proposal
pi_prime = np.random.dirichlet(beta * pi)
n_total += 1
### get proposal probability and sample it
new_val = log_censored_dirichlet_density(pi_prime,alpha=alpha,data=data)
if new_val > -np.inf: # in our tests, this is always true
a = min(1.,np.exp(new_val - current_val
+ log_dirichlet_density(pi,alpha=beta*pi_prime)
- log_dirichlet_density(pi_prime,alpha=beta*pi)))
if np.random.rand() < a:
n_accepts += 1
pi = pi_prime
current_val = new_val
samples.append(pi)
# print 'done drawing samples in %0.2f seconds' % (time.time() - starttime)
# print '%d total proposals, %d valid proposals, %d accepted, valid acceptance ratio %0.4f' % (n_total,n_samples, n_accepts, n_accepts / n_samples)
return samples
def prior_posterior_2D(meshsize=250,alpha=2.,data=np.array([[0,2,0],[0,0,0],[0,0,0]])):
assert data.shape == (3,3)
mesh3D = simplex.mesh(meshsize)
mesh2D = simplex.proj_to_2D(mesh3D) # use specialized b/c it plays nicer with triangulation algorithm
priorvals = np.exp(dirichlet.log_dirichlet_density(mesh3D,alpha))
posteriorvals_uncensored = np.exp(dirichlet.log_dirichlet_density(mesh3D,alpha,data=data.sum(0)))
temp = dirichlet.log_censored_dirichlet_density(mesh3D,alpha,data=data)
temp = np.exp(temp - temp.max())
posteriorvals_censored = temp/temp.sum() # direct discretized integration!
# used for grid interpolation
xi = np.linspace(mesh2D[:,0].min(), mesh2D[:,0].max(), 2000, endpoint=True)
yi = np.linspace(mesh2D[:,1].min(), mesh2D[:,1].max(), 2000, endpoint=True)
plt.figure(figsize=(8,8))
# use exactly one of the next two code lines!
# this one performs interpolation to get a rectangular-pixel grid, but
# produces a blurred image
plt.imshow(griddata((mesh2D[:,0],mesh2D[:,1]),priorvals,(xi[na,:],yi[:,na]),method='cubic'))
# this one exactly represents the data by performing a DeLaunay
# triangulation, but it must draw each triangular pixel individually,
# resulting in large files and slow draw times
# plt.tripcolor(mesh2D[:,0],mesh2D[:,1],priorvals) # exact triangles, no blurring
plt.axis('off')
save('../writeup/figures/dirichlet_prior_2D.pdf')
plt.figure(figsize=(8,8))
plt.imshow(griddata((mesh2D[:,0],mesh2D[:,1]),posteriorvals_uncensored,(xi[na,:],yi[:,na]),method='cubic'))
# plt.tripcolor(mesh2D[:,0],mesh2D[:,1],posteriorvals_uncensored)
plt.axis('off')
save('../writeup/figures/dirichlet_uncensored_posterior_2D.pdf')
plt.figure(figsize=(8,8))
plt.imshow(griddata((mesh2D[:,0],mesh2D[:,1]),posteriorvals_censored,(xi[na,:],yi[:,na]),method='cubic'))
# plt.tripcolor(mesh2D[:,0],mesh2D[:,1],posteriorvals_censored)
plt.axis('off')
save('../writeup/figures/dirichlet_censored_posterior_2D.pdf')