# step = pm.NUTS() # Instantiate NUTS sampler
# trace = pm.sample(5000, step, start=start, progressbar=False)
# create an array with the posterior sample
theta_sample = trace['theta']
print theta_sample
plt.subplot(1, 2, 1)
plt.plot(theta_sample[:500], np.arange(500), marker='o')
plt.xlim(0, 1)
plt.xlabel(r'$\theta$')
plt.ylabel('Position in Chain')
plt.subplot(1, 2, 2)
mcmc_info = plot_post(theta_sample, xlab=r'$\theta', show_mode=False)
# Posterior prediction:
# For each step in the chain, use posterior theta to flip a coin:
y_pred = np.zeros(len(theta_sample))
for i, p_head in enumerate(theta_sample):
y_pred[i] = np.random.choice([0, 1], p=[1 - p_head, p_head])
# Jitter the 0,1 y values for plotting purposes:
y_pred_jittered = y_pred + np.random.uniform(-.05, .05, size=len(theta_sample))
# Now plot the jittered values:
plt.figure()
plt.plot(theta_sample[:500], y_pred_jittered[:500], 'ro')
plt.xlim(-.1, 1.1)
plt.ylim(-.1, 1.1)
# step = pm.NUTS() # Instantiate NUTS sampler
# trace = pm.sample(5000, step, start=start, progressbar=False)
# create an array with the posterior sample
theta_sample = trace['theta']
print theta_sample
plt.subplot(1, 2, 1)
plt.plot(theta_sample[:500], np.arange(500), marker='o')
plt.xlim(0, 1)
plt.xlabel(r'$\theta$')
plt.ylabel('Position in Chain')
plt.subplot(1, 2, 2)
mcmc_info = plot_post(theta_sample, xlab=r'$\theta', show_mode=False)
# Posterior prediction:
# For each step in the chain, use posterior theta to flip a coin:
y_pred = np.zeros(len(theta_sample))
for i, p_head in enumerate(theta_sample):
y_pred[i] = np.random.choice([0, 1], p=[1 - p_head, p_head])
# Jitter the 0,1 y values for plotting purposes:
y_pred_jittered = y_pred + np.random.uniform(-.05, .05, size=len(theta_sample))
# Now plot the jittered values:
plt.figure()
plt.plot(theta_sample[:500], y_pred_jittered[:500], 'ro')
plt.xlim(-.1, 1.1)
plt.ylim(-.1, 1.1)
# Evidence for model, p(D).
# Compute a,b parameters for beta distribution that has the same mean
# and stdev as the sample from the posterior. This is a useful choice
# when the likelihood function is Bernoulli.
a = mean_traj * ((mean_traj*(1 - mean_traj)/std_traj**2) - 1)
b = (1 - mean_traj) * ((mean_traj*(1 - mean_traj)/std_traj**2) - 1)
# For every theta value in the posterior sample, compute
# dbeta(theta,a,b) / likelihood(theta)*prior(theta)
# This computation assumes that likelihood and prior are proper densities,
# i.e., not just relative probabilities. This computation also assumes that
# the likelihood and prior functions were defined to accept a vector argument,
# not just a single-component scalar argument.
wtd_evid = beta.pdf(accepted_traj, a, b) / (likelihood(accepted_traj, my_data) * prior(accepted_traj))
p_data = 1 / np.mean(wtd_evid)
# Display p(D) in the graph
plt.plot(0, label='p(D) = %.3e' % p_data, alpha=0)
# Display the posterior.
ROPE = np.array([0.76, 0.8])
mcmc_info = plot_post(accepted_traj, xlab='theta', show_mode=False, comp_val=0.9, ROPE=ROPE)
# Uncomment next line if you want to save the graph.
plt.savefig('BernMetropolisTemplate.png')
plt.show()
