def plot_post(param_sample_vec, cred_mass=0.95, comp_val=False,
ROPE=False, ylab='', xlab='parameter', fontsize=14, labelsize=14,
title='', framealpha=1, facecolor='skyblue', edgecolor='white',
show_mode=True, bins=50):
#compute HDI
HDI = hpd(param_sample_vec, 1-cred_mass)
post_summary = {'mean':0,'median':0,'mode':0, 'hdi_mass':0,'hdi_low':0,
'hdi_high':0, 'comp_val':0, 'pc_gt_comp_val':0, 'ROPE_low':0,
'ROPE_high':0, 'pc_in_ROPE':0}
post_summary['mean'] = np.mean(param_sample_vec)
post_summary['median'] = np.median(param_sample_vec)
post_summary['mode'] = stats.mode(param_sample_vec)[0]
post_summary['hdi_mass'] = cred_mass
post_summary['hdi_low'] = HDI[0]
post_summary['hdi_high'] = HDI[1]
# Plot histogram.
n, bins, patches = plt.hist(param_sample_vec, normed=True, bins=bins,
edgecolor=edgecolor, facecolor=facecolor)
plt.xlabel(xlab, fontsize=fontsize)
plt.ylabel(ylab, fontsize=fontsize)
plt.title(title, fontsize=fontsize)
cv_ht = 0.75*np.max(n)
cen_tend_ht = 0.9 * cv_ht
ROPE_text_ht = 0.55 * cv_ht
# # Display mean or mode:
if show_mode:
plt.plot(0, label='mode = %.2f' % post_summary['mode'], alpha=0)
else:
plt.plot(0, label='mean = %.2f' % post_summary['mean'], alpha=0)
# Display the comparison value.
if comp_val is not False:
pc_gt_comp_val = 100 * np.sum(param_sample_vec > comp_val)/len(param_sample_vec)
pc_lt_comp_val = 100 - pc_gt_comp_val
plt.plot([comp_val, comp_val], [0, cv_ht], color='darkgreen',
linestyle='--', linewidth=2,
label='%.1f%% <%.1f < %.1f%%'
% (pc_lt_comp_val, comp_val, pc_gt_comp_val))
post_summary['comp_val'] = comp_val
post_summary['pc_gt_comp_val'] = pc_gt_comp_val
# # Display the ROPE.
if ROPE is not False:
rope_col = 'darkred'
pc_in_ROPE = round(np.sum((param_sample_vec > ROPE[0]) & (param_sample_vec < ROPE[1]))/len(param_sample_vec)*100)
plt.plot([ROPE[0], ROPE[0]], [0, 0.96*ROPE_text_ht], color=rope_col,
linestyle=':', linewidth=4,
label='%.1f%% in ROPE' % pc_in_ROPE)
plt.plot([ROPE[1], ROPE[1]], [0, 0.96*ROPE_text_ht], color=rope_col,
linestyle=':', linewidth=4)
post_summary['ROPE_low'] = ROPE[0]
post_summary['ROPE_high'] = ROPE[1]
post_summary['pc_in_ROPE'] = pc_in_ROPE
# # Display the HDI.
plt.plot(HDI, [0, 0], linewidth=6, color='k', label='HDI %.1f%% %.3f-%.3f' % (cred_mass*100, HDI[0], HDI[1]))
plt.legend(loc='upper left', fontsize=labelsize, framealpha=framealpha)
return post_summary
def bern_grid(theta, p_theta, data, credib=.95):
"""
Bayesian updating for Bernoulli likelihood and prior specified on a grid.
Input arguments:
theta is a vector of theta values, all between 0 and 1.
p_theta is a vector of corresponding probability _masses_.
data is a vector of 1's and 0's, where 1 corresponds to a and 0 to b.
credib is the probability mass of the credible interval, default is 0.95.
Output:
p_theta_given_data is a vector of posterior probability masses over theta.
Also creates a three-panel graph of prior, likelihood, and posterior
probability masses with credible interval.
Example of use:
Create vector of theta values.
bin_width = 1/1000
theta_grid = np.arange(0, 1+bin_width, bin_width)
Specify probability mass at each theta value.
> rel_prob = np.minimum(theta_grid, 1-theta_grid) relative prob at each theta
> prior = rel_prob / sum(rel_prob) probability mass at each theta
Specify the data vector.
data_vec = np.repeat([1, 0], [11, 3]) # 3 heads, 1 tail
Call the function.
> posterior = bern_grid( theta=theta_grid , p_theta=prior , data=data_vec )
"""
import matplotlib.pyplot as plt
import numpy as np
from hpd import hpd
# Create summary values of data
z = sum(data[data == 1]) # number of 1's in data
N = len(data) # number of flips in data
# Compute the likelihood of the data for each value of theta.
p_data_given_theta = theta**z * (1 - theta)**(N - z)
# Compute the evidence and the posterior.
p_data = sum(p_data_given_theta * p_theta)
p_theta_given_data = p_data_given_theta * p_theta / p_data
# Determine the limits of the highest density interval
x = np.random.choice(theta, size=5000, replace=True, p=p_theta_given_data)
intervals = hpd(x, alpha=1-credib)
# Plot the results.
plt.figure(figsize=(12, 12))
plt.subplots_adjust(hspace=0.7)
# # Plot the prior.
locx = 0.05
mean_theta = sum(theta * p_theta) # mean of prior, for plotting
plt.subplot(3, 1, 1)
plt.plot(theta, p_theta)
plt.xlim(0, 1)
plt.ylim(0, np.max(p_theta)*1.2)
plt.xlabel(r'$\theta$')
plt.ylabel(r'$P(\theta)$')
plt.title('Prior')
plt.text(locx, np.max(p_theta)/2, r'mean($\theta$;%5.2f)' % mean_theta)
# Plot the likelihood:
plt.subplot(3, 1, 2)
plt.plot(theta, p_data_given_theta)
plt.xlim(0, 1)
plt.ylim(0, np.max(p_data_given_theta)*1.2)
plt.xlabel(r'$\theta$')
plt.ylabel(r'$P(D|\theta)$')
plt.title('Likelihood')
plt.text(locx, np.max(p_data_given_theta)/2, 'data: z=%s, N=%s' % (z, N))
# Plot the posterior:
mean_theta_given_data = sum(theta * p_theta_given_data)
plt.subplot(3, 1, 3)
plt.plot(theta, p_theta_given_data)
plt.xlim(0, 1)
plt.ylim(0, np.max(p_theta_given_data)*1.2)
plt.xlabel(r'$\theta$')
plt.ylabel(r'$P(\theta|D)$')
plt.title('Posterior')
loc = np.linspace(0, np.max(p_theta_given_data), 5)
plt.text(locx, loc[1], r'mean($\theta$;%5.2f)' % mean_theta_given_data)
plt.text(locx, loc[2], 'P(D) = %g' % p_data)
# Plot the HDI
plt.text(locx, loc[3],
'Intervals =%s' % ', '.join('%.3f' % x for x in intervals))
for i in range(0, len(intervals), 2):
plt.fill_between(theta, 0, p_theta_given_data,
where=np.logical_and(theta > intervals[i],
theta < intervals[i+1]),
color='blue', alpha=0.3)
plt.savefig('Figure_6.1.png')
plt.show()
return p_theta_given_data
def plot_post(param_sample_vec,
cred_mass=0.95,
comp_val=False,
ROPE=False,
ylab='',
xlab='parameter',
fontsize=14,
labelsize=14,
title='',
framealpha=1,
facecolor='skyblue',
edgecolor='white',
show_mode=True,
bins=50):
#compute HDI
HDI = hpd(param_sample_vec, 1 - cred_mass)
post_summary = {
'mean': 0,
'median': 0,
'mode': 0,
'hdi_mass': 0,
'hdi_low': 0,
'hdi_high': 0,
'comp_val': 0,
'pc_gt_comp_val': 0,
'ROPE_low': 0,
'ROPE_high': 0,
'pc_in_ROPE': 0
}
post_summary['mean'] = np.mean(param_sample_vec)
post_summary['median'] = np.median(param_sample_vec)
post_summary['mode'] = stats.mode(param_sample_vec)[0]
post_summary['hdi_mass'] = cred_mass
post_summary['hdi_low'] = HDI[0]
post_summary['hdi_high'] = HDI[1]
# Plot histogram.
n, bins, patches = plt.hist(param_sample_vec,
normed=True,
bins=bins,
edgecolor=edgecolor,
facecolor=facecolor)
plt.xlabel(xlab, fontsize=fontsize)
plt.ylabel(ylab, fontsize=fontsize)
plt.title(title, fontsize=fontsize)
cv_ht = 0.75 * np.max(n)
cen_tend_ht = 0.9 * cv_ht
ROPE_text_ht = 0.55 * cv_ht
# # Display mean or mode:
if show_mode:
plt.plot(0, label='mode = %.2f' % post_summary['mode'], alpha=0)
else:
plt.plot(0, label='mean = %.2f' % post_summary['mean'], alpha=0)
# Display the comparison value.
if comp_val is not False:
pc_gt_comp_val = 100 * np.sum(
param_sample_vec > comp_val) / len(param_sample_vec)
pc_lt_comp_val = 100 - pc_gt_comp_val
plt.plot([comp_val, comp_val], [0, cv_ht],
color='darkgreen',
linestyle='--',
linewidth=2,
label='%.1f%% <%.1f < %.1f%%' %
(pc_lt_comp_val, comp_val, pc_gt_comp_val))
post_summary['comp_val'] = comp_val
post_summary['pc_gt_comp_val'] = pc_gt_comp_val
# # Display the ROPE.
if ROPE is not False:
rope_col = 'darkred'
pc_in_ROPE = round(
np.sum((param_sample_vec > ROPE[0]) & (param_sample_vec < ROPE[1]))
/ len(param_sample_vec) * 100)
plt.plot([ROPE[0], ROPE[0]], [0, 0.96 * ROPE_text_ht],
color=rope_col,
linestyle=':',
linewidth=4,
label='%.1f%% in ROPE' % pc_in_ROPE)
plt.plot([ROPE[1], ROPE[1]], [0, 0.96 * ROPE_text_ht],
color=rope_col,
linestyle=':',
linewidth=4)
post_summary['ROPE_low'] = ROPE[0]
post_summary['ROPE_high'] = ROPE[1]
post_summary['pc_in_ROPE'] = pc_in_ROPE
# # Display the HDI.
plt.plot(HDI, [0, 0],
linewidth=6,
color='k',
label='HDI %.1f%% %.3f-%.3f' % (cred_mass * 100, HDI[0], HDI[1]))
plt.legend(loc='upper left', fontsize=labelsize, framealpha=framealpha)
frame = plt.gca()
frame.axes.get_yaxis().set_ticks([])
return post_summary
def bern_grid(theta, p_theta, data, credib=.95):
"""
Bayesian updating for Bernoulli likelihood and prior specified on a grid.
Input arguments:
theta is a vector of theta values, all between 0 and 1.
p_theta is a vector of corresponding probability _masses_.
data is a vector of 1's and 0's, where 1 corresponds to a and 0 to b.
credib is the probability mass of the credible interval, default is 0.95.
Output:
p_theta_given_data is a vector of posterior probability masses over theta.
Also creates a three-panel graph of prior, likelihood, and posterior
probability masses with credible interval.
Example of use:
Create vector of theta values.
bin_width = 1/1000
theta_grid = np.arange(0, 1+bin_width, bin_width)
Specify probability mass at each theta value.
> rel_prob = np.minimum(theta_grid, 1-theta_grid) relative prob at each theta
> prior = rel_prob / sum(rel_prob) probability mass at each theta
Specify the data vector.
data_vec = np.repeat([1, 0], [11, 3]) # 3 heads, 1 tail
Call the function.
> posterior = bern_grid( theta=theta_grid , p_theta=prior , data=data_vec )
"""
# Create summary values of data
z = sum(data[data == 1]) # number of 1's in data
N = len(data) # number of flips in data
# Compute the likelihood of the data for each value of theta.
p_data_given_theta = theta**z * (1 - theta)**(N - z)
# Compute the evidence and the posterior.
p_data = sum(p_data_given_theta * p_theta)
p_theta_given_data = p_data_given_theta * p_theta / p_data
# Determine the limits of the highest density interval
x = np.random.choice(theta, size=5000, replace=True, p=p_theta_given_data)
intervals = hpd(x, alpha=1 - credib)
# Plot the results.
plt.figure(figsize=(12, 12))
plt.subplots_adjust(hspace=0.7)
# # Plot the prior.
locx = 0.05
mean_theta = sum(theta * p_theta) # mean of prior, for plotting
plt.subplot(3, 1, 1)
plt.plot(theta, p_theta)
plt.xlim(0, 1)
plt.ylim(0, np.max(p_theta) * 1.2)
plt.xlabel(r'$\theta$')
plt.ylabel(r'$P(\theta)$')
plt.title('Prior')
plt.text(locx, np.max(p_theta) / 2, r'mean($\theta$;%5.2f)' % mean_theta)
# Plot the likelihood:
plt.subplot(3, 1, 2)
plt.plot(theta, p_data_given_theta)
plt.xlim(0, 1)
plt.ylim(0, np.max(p_data_given_theta) * 1.2)
plt.xlabel(r'$\theta$')
plt.ylabel(r'$P(D|\theta)$')
plt.title('Likelihood')
plt.text(locx, np.max(p_data_given_theta) / 2, 'data: z=%s, N=%s' % (z, N))
# Plot the posterior:
mean_theta_given_data = sum(theta * p_theta_given_data)
plt.subplot(3, 1, 3)
plt.plot(theta, p_theta_given_data)
plt.xlim(0, 1)
plt.ylim(0, np.max(p_theta_given_data) * 1.2)
plt.xlabel(r'$\theta$')
plt.ylabel(r'$P(\theta|D)$')
plt.title('Posterior')
loc = np.linspace(0, np.max(p_theta_given_data), 5)
plt.text(locx, loc[1], r'mean($\theta$;%5.2f)' % mean_theta_given_data)
plt.text(locx, loc[2], 'P(D) = %g' % p_data)
# Plot the HDI
plt.text(locx, loc[3],
'Intervals =%s' % ', '.join('%.3f' % x for x in intervals))
for i in range(0, len(intervals), 2):
plt.fill_between(theta,
0,
p_theta_given_data,
where=np.logical_and(theta > intervals[i],
theta < intervals[i + 1]),
color='blue',
alpha=0.3)
plt.savefig('Figure_6.1.png')
plt.show()
return p_theta_given_data
def plot_post(
param_sample_vec,
cred_mass=0.95,
comp_val=False,
ROPE=False,
ylab="",
xlab="parameter",
fontsize=14,
title="",
facecolor="skyblue",
edgecolor="white",
show_mode=True,
bins=50,
):
# compute HDI
HDI = hpd(param_sample_vec, 1 - cred_mass)
post_summary = {
"mean": 0,
"median": 0,
"mode": 0,
"hdi_mass": 0,
"hdi_low": 0,
"hdi_high": 0,
"comp_val": 0,
"pc_gt_comp_val": 0,
"ROPE_low": 0,
"ROPE_high": 0,
"pc_in_ROPE": 0,
}
post_summary["mean"] = np.mean(param_sample_vec)
post_summary["median"] = np.median(param_sample_vec)
post_summary["mode"] = stats.mode(param_sample_vec)[0]
post_summary["hdi_mass"] = cred_mass
post_summary["hdi_low"] = HDI[0]
post_summary["hdi_high"] = HDI[1]
# Plot histogram.
n, bins, patches = plt.hist(param_sample_vec, normed=True, bins=bins, edgecolor=edgecolor, facecolor=facecolor)
plt.xlabel(xlab, fontsize=fontsize)
plt.ylabel(ylab, fontsize=fontsize)
plt.title(title, fontsize=fontsize)
idx = (np.abs(bins - comp_val)).argmin() # index of the nearest value to comp_val
cvHt = n[idx - 1]
cenTendHt = 0.9 * cvHt
ROPEtextHt = 0.55 * cvHt
# # Display mean or mode:
if show_mode:
plt.plot(0, label="mode = %.2f" % post_summary["mode"], alpha=0)
else:
plt.plot(0, label="mean = %.2f" % post_summary["mean"], alpha=0)
# Display the comparison value.
if comp_val is not False:
pc_gt_comp_val = 100 * np.sum(param_sample_vec > comp_val) / len(param_sample_vec)
pc_lt_comp_val = 100 - pc_gt_comp_val
plt.plot(
[comp_val, comp_val],
[0, cvHt],
color="darkgreen",
linestyle="--",
linewidth=2,
label="%.1f%% <%.1f < %.1f%%" % (pc_lt_comp_val, comp_val, pc_gt_comp_val),
)
post_summary["comp_val"] = comp_val
post_summary["pc_gt_comp_val"] = pc_gt_comp_val
# # Display the ROPE.
if ROPE is not False:
rope_col = "darkred"
pc_in_ROPE = round(
np.sum((param_sample_vec > ROPE[0]) & (param_sample_vec < ROPE[1])) / len(param_sample_vec) * 100
)
plt.plot(
[ROPE[0], ROPE[0]],
[0, 0.96 * ROPEtextHt],
color=rope_col,
linestyle=":",
linewidth=4,
label="%.1f%% in ROPE" % pc_in_ROPE,
)
plt.plot([ROPE[1], ROPE[1]], [0, 0.96 * ROPEtextHt], color=rope_col, linestyle=":", linewidth=4)
post_summary["ROPE_low"] = ROPE[0]
post_summary["ROPE_high"] = ROPE[1]
post_summary["pc_in_ROPE"] = pc_in_ROPE
# # Display the HDI.
plt.plot(HDI, [0, 0], linewidth=6, color="k", label="HDI %.1f%% %.3f-%.3f" % (cred_mass * 100, HDI[0], HDI[1]))
plt.legend(loc="upper left")
return post_summary
# locyp, locint = [], [] # for ttt in tt: # locyp.append(xtest) # locint.append(pdftest) # yp.append(locyp) # integrall.append(locint) ############################################################################### # Compute credibility intervals for each experiment: beta = [] for x_L, pdf_L, obs_L in zip(allYP, allINTEGRAL, allh): beta_loc = [] for x, pdf, obs in zip(x_L, pdf_L, obs_L): beta_loc.append( hpd(x, pdf, obs) ) beta.append(beta_loc) print '# betas!' for i in xrange(len(beta[0])-1): #print allYP[i], beta[i] print times[i], 1-beta[1][i] ############################################################################### # Plot posterior for obs along with obs for a few cases # and save it to a file Obs_ind = [5, 20, 35] for beta_loc, filename, x_L, pdf_L, obs_L in zip(beta, DataFilesNames, allYP, allINTEGRAL, allh): justnameoffile, extension = os.path.splitext(filename) maxind = np.where(beta_loc == max(beta_loc))[0][0]
def bern_beta(prior_shape, data_vec, cred_mass=0.95):
"""Bayesian updating for Bernoulli likelihood and beta prior.
Input arguments:
prior_shape
vector of parameter values for the prior beta distribution.
data_vec
vector of 1's and 0's.
cred_mass
the probability mass of the HDI.
Output:
post_shape
vector of parameter values for the posterior beta distribution.
Graphics:
Creates a three-panel graph of prior, likelihood, and posterior
with highest posterior density interval.
Example of use:
post_shape = bern_beta(prior_shape=[1,1] , data_vec=[1,0,0,1,1])"""
# Check for errors in input arguments:
if len(prior_shape) != 2:
sys.exit('prior_shape must have two components.')
if any([i < 0 for i in prior_shape]):
sys.exit('prior_shape components must be positive.')
if any([i != 0 and i != 1 for i in data_vec]):
sys.exit('data_vec must be a vector of 1s and 0s.')
if cred_mass <= 0 or cred_mass >= 1.0:
sys.exit('cred_mass must be between 0 and 1.')
# Rename the prior shape parameters, for convenience:
a = prior_shape[0]
b = prior_shape[1]
# Create summary values of the data:
z = sum(data_vec[data_vec == 1]) # number of 1's in data_vec
N = len(data_vec) # number of flips in data_vec
# Compute the posterior shape parameters:
post_shape = [a+z, b+N-z]
# Compute the evidence, p(D):
p_data = beta_func(z+a, N-z+b)/beta_func(a, b)
# Construct grid of theta values, used for graphing.
bin_width = 0.005 # Arbitrary small value for comb on theta.
theta = np.arange(bin_width/2, 1-(bin_width/2)+bin_width, bin_width)
# Compute the prior at each value of theta.
p_theta = beta.pdf(theta, a, b)
# Compute the likelihood of the data at each value of theta.
p_data_given_theta = theta**z * (1-theta)**(N-z)
# Compute the posterior at each value of theta.
post_a = a + z
post_b = b+N-z
p_theta_given_data = beta.pdf(theta, a+z, b+N-z)
# Determine the limits of the highest density interval
x = np.random.beta(a=post_shape[0], b=post_shape[1], size=5000)
intervals = hpd(x, alpha=1-cred_mass) # hdp is a function from PyMC package
# takes a sample vector as input, not
# a function, like the HDIofICDF.R
# Plot the results.
plt.figure(figsize=(12, 12))
plt.subplots_adjust(hspace=0.7)
# Plot the prior.
locx = 0.05
plt.subplot(3, 1, 1)
plt.plot(theta, p_theta)
plt.xlim(0, 1)
plt.ylim(0, np.max(p_theta)*1.2)
plt.xlabel('$\\theta$')
plt.ylabel('$P(\\theta)$')
plt.title('Prior')
plt.text(locx, np.max(p_theta)/2, 'beta($\\theta$;%s,%s)' % (a, b))
# Plot the likelihood:
plt.subplot(3, 1, 2)
plt.plot(theta, p_data_given_theta)
plt.xlim(0, 1)
plt.ylim(0, np.max(p_data_given_theta)*1.2)
plt.xlabel('$\\theta$')
plt.ylabel('$P(D|\\theta)$')
plt.title('Likelihood')
plt.text(locx, np.max(p_data_given_theta)/2, 'Data: z=%s, N=%s' % (z, N))
# Plot the posterior:
plt.subplot(3, 1, 3)
plt.plot(theta, p_theta_given_data)
plt.xlim(0, 1)
plt.ylim(0, np.max(p_theta_given_data)*1.2)
plt.xlabel('$\\theta$')
plt.ylabel('$P(\\theta|D)$')
plt.title('Posterior')
locy = np.linspace(0, np.max(p_theta_given_data), 5)
plt.text(locx, locy[1], 'beta($\\theta$;%s,%s)' % (post_a, post_b))
plt.text(locx, locy[2], 'P(D) = %g' % p_data)
# Plot the HDI
plt.text(locx, locy[3],
'Intervals =%s' % ', '.join('%.3f' % x for x in intervals))
for i in range(0, len(intervals), 2):
plt.fill_between(theta, 0, p_theta_given_data,
where=np.logical_and(theta > intervals[i],
theta < intervals[i+1]),
color='blue', alpha=0.3)
return intervals
def plot_post(param_sample_vec, cred_mass=0.95, comp_val=None,
ROPE=None, ylab='', xlab='parameter', fontsize=14, labelsize=14,
title='', framealpha=1, show_mode=True, bins=50, kde_plot=True,
roundto=3):
"""
Write me!
"""
# colors taken from the default seaborn color pallete
blue, green, red = [(0.2980392156862745, 0.4470588235294118,
0.6901960784313725), (0.3333333333333333, 0.6588235294117647,
0.40784313725490196), (0.7686274509803922, 0.3058823529411765,
0.3215686274509804)]
## Compute HDI
HDI = hpd(param_sample_vec, 1-cred_mass)
post_summary = {'mean':0,'median':0,'mode':0, 'hdi_mass':0,'hdi_low':0,
'hdi_high':0, 'comp_val':0, 'pc_gt_comp_val':0, 'ROPE_low':0,
'ROPE_high':0, 'pc_in_ROPE':0}
post_summary['mean'] = round(np.mean(param_sample_vec), roundto)
post_summary['median'] = round(np.median(param_sample_vec), roundto)
post_summary['mode'] = round(stats.mode(param_sample_vec)[0], roundto)
post_summary['hdi_mass'] = cred_mass
post_summary['hdi_low'] = round(HDI[0], roundto)
post_summary['hdi_high'] = round(HDI[1], roundto)
## Plot KDE.
if kde_plot:
density = stats.kde.gaussian_kde(param_sample_vec)
l = np.min(param_sample_vec)
u = np.max(param_sample_vec)
x = np.linspace(0, 1, 100) * (u - l) + l
plt.plot(x, density(x), color=blue)
## Plot histogram.
else:
plt.hist(param_sample_vec, normed=True, bins=bins, facecolor=blue,
edgecolor='w')
## Display mean or mode:
if show_mode:
plt.plot(0, label='mode = %.2f' % post_summary['mode'], alpha=0)
else:
plt.plot(0, label='mean = %.2f' % post_summary['mean'], alpha=0)
## Display the comparison value.
if comp_val is not None:
pc_gt_comp_val = 100 * np.sum(param_sample_vec > comp_val)/len(param_sample_vec)
pc_lt_comp_val = 100 - pc_gt_comp_val
plt.axvline(comp_val, ymax=.75, color=green,
linestyle='--', linewidth=4,
label='%.1f%% < %.1f < %.1f%%'
% (pc_lt_comp_val, comp_val, pc_gt_comp_val))
post_summary['comp_val'] = comp_val
post_summary['pc_gt_comp_val'] = pc_gt_comp_val
## Display the ROPE.
if ROPE is not None:
rope_col = 'darkred'
pc_in_ROPE = round(np.sum((param_sample_vec > ROPE[0]) & (param_sample_vec < ROPE[1]))/len(param_sample_vec)*100)
plt.axvline(ROPE[0], ymax=.75, color=red, linewidth=4,
label='%.1f%% in ROPE' % pc_in_ROPE)
plt.axvline(ROPE[1], ymax=.75, color=red, linewidth=4)
post_summary['ROPE_low'] = ROPE[0]
post_summary['ROPE_high'] = ROPE[1]
post_summary['pc_in_ROPE'] = pc_in_ROPE
## Display the HDI.
plt.plot(HDI, [0, 0], linewidth=8, color='k', label='HDI %.1f%% %.3f-%.3f' % (cred_mass*100, HDI[0], HDI[1]))
plt.xlabel(xlab, fontsize=fontsize)
plt.ylabel(ylab, fontsize=fontsize)
plt.title(title, fontsize=fontsize)
plt.legend(loc='upper left', fontsize=labelsize, framealpha=framealpha)
frame = plt.gca()
frame.axes.get_yaxis().set_ticks([])
return post_summary
