In [1]:
# %load std_ipython_import.txt
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
import pymc3 as pm
import theano.tensor as tt
import warnings
warnings.filterwarnings("ignore", category=FutureWarning)
from theano.compile.ops import as_op
from scipy.stats import norm
from matplotlib import gridspec
from matplotlib.patches import Rectangle
from IPython.display import Image
%matplotlib inline
plt.style.use('seaborn-white')
color = '#87ceeb'
f_dict = {'size':14}
In [2]:
%load_ext watermark
%watermark -p pandas,numpy,pymc3,theano,matplotlib,seaborn,scipy
Code based on https://gist.github.com/DanielWeitzenfeld/d9ac64f76281e6c1d29217af76449664
In [3]:
# Using dtype 'category' for Y
df = pd.read_csv('data/OrdinalProbitData-1grp-1.csv', dtype={'Y':'category'})
df.info()
In [4]:
df.Y.value_counts(sort=False)
Out[4]:
In [5]:
# Number of outcomes
nYlevels = df.Y.cat.categories.size
thresh = np.arange(1.5, nYlevels, dtype=np.float32)
thresh_obs = np.ma.asarray(thresh)
thresh_obs[1:-1] = np.ma.masked
print('thresh:\t\t{}'.format(thresh))
print('thresh_obs:\t{}'.format(thresh_obs))
In [6]:
# Using the Theano @as_op decorator with a custom function to calculate the threshold probabilities.
# Theano cannot compute a gradient for these custom functions, so it is not possible to use
# gradient based samplers in PyMC3.
# http://pymc-devs.github.io/pymc3/notebooks/getting_started.html#Arbitrary-deterministics
@as_op(itypes=[tt.fvector, tt.fscalar, tt.fscalar], otypes=[tt.fvector])
def outcome_probabilities(theta, mu, sigma):
out = np.empty(nYlevels, dtype=np.float32)
n = norm(loc=mu, scale=sigma)
out[0] = n.cdf(theta[0])
out[1] = np.max([0, n.cdf(theta[1]) - n.cdf(theta[0])])
out[2] = np.max([0, n.cdf(theta[2]) - n.cdf(theta[1])])
out[3] = np.max([0, n.cdf(theta[3]) - n.cdf(theta[2])])
out[4] = np.max([0, n.cdf(theta[4]) - n.cdf(theta[3])])
out[5] = np.max([0, n.cdf(theta[5]) - n.cdf(theta[4])])
out[6] = 1 - n.cdf(theta[5])
return out
with pm.Model() as ordinal_model_single:
theta = pm.Normal('theta', mu=thresh, tau=np.repeat(.5**2, len(thresh)),
shape=len(thresh), observed=thresh_obs)
mu = pm.Normal('mu', mu=nYlevels/2.0, tau=1.0/(nYlevels**2))
sigma = pm.Uniform('sigma', nYlevels/1000.0, nYlevels*10.0)
pr = outcome_probabilities(theta, mu, sigma)
y = pm.Categorical('y', pr, observed=df.Y.cat.codes.values)
pm.model_to_graphviz(ordinal_model_single)
Out[6]:
In [7]:
with ordinal_model_single:
trace1 = pm.sample(3000, cores=4)
In [8]:
pm.traceplot(trace1);
In [9]:
mu = trace1['mu']
sigma = trace1['sigma']
# Concatenate the fixed thresholds into the estimated thresholds
n = trace1['theta_missing'].shape[0]
thresholds = np.c_[np.tile([1.5], (n,1)),
trace1['theta_missing'],
np.tile([6.5], (n,1))]
# Define gridspec
fig = plt.figure(figsize=(10,8))
gs = gridspec.GridSpec(3, 2)
ax1 = plt.subplot(gs[0,0])
ax2 = plt.subplot(gs[0,1])
ax3 = plt.subplot(gs[1,0])
ax4 = plt.subplot(gs[1,1])
ax5 = plt.subplot(gs[2,0])
# Mu
pm.plot_posterior(mu, point_estimate='mode', color=color, ax=ax1)
ax1.set_title('Mean', fontdict=f_dict)
ax1.set_xlabel('$\mu$', fontdict=f_dict)
# Posterior predictive probabilities of the outcomes
threshCumProb = np.empty(thresholds.shape)
for i in np.arange(threshCumProb.shape[0]):
threshCumProb[i] = norm().cdf((thresholds[i] - mu[i])/sigma[i])
outProb = (np.c_[threshCumProb, np.tile(1, (thresholds.shape[0],1))]
- np.c_[np.tile(0, (thresholds.shape[0],1)), threshCumProb])
yerr = np.abs(np.subtract(pm.hpd(outProb), outProb.mean(axis=0).reshape(-1,1)))
(df.Y.value_counts()/df.Y.size).plot.bar(ax=ax2, rot=0, color='royalblue')
ax2.errorbar(x = np.arange(df.Y.nunique()), y=outProb.mean(axis=0),
yerr=yerr.T, color=color, fmt='o')
ax2.set_xlabel('y')
sns.despine(ax=ax2, left=True)
ax2.yaxis.set_visible(False)
ax2.set_title('Data w. Post. Pred.\n N={}'.format(df.Y.size), fontdict=f_dict)
# Sigma
pm.plot_posterior(sigma, point_estimate='mode', color=color, ax=ax3)
ax3.set_title('Std. Dev.', fontdict=f_dict)
ax3.set_xlabel('$\sigma$', fontdict=f_dict)
# Effect size
pm.plot_posterior((mu-2)/sigma,point_estimate='mode', color=color, ax=ax4)
ax4.set_title('Effect Size', fontdict=f_dict)
ax4.set_xlabel('$(\mu-2)/\sigma$', fontdict=f_dict)
# Posterior distribution on the thresholds
ax5.scatter(thresholds, np.tile(thresholds.mean(axis=1).reshape(-1,1), (1,6)), color=color, alpha=.6, facecolor='none')
ax5.set_ylabel('Mean Threshold', fontdict=f_dict)
ax5.set_xlabel('Threshold', fontdict=f_dict)
ax5.vlines(x = thresholds.mean(axis=0),
ymin=thresholds.mean(axis=1).min(),
ymax=thresholds.mean(axis=1).max(), linestyles='dotted', colors=color)
fig.tight_layout()
In [10]:
# Using dtype 'category' for X & Y
df2 = pd.read_csv('data/OrdinalProbitData1.csv', dtype={'X':'category','Y':'category'})
df2.info()
In [11]:
sns.countplot(x=df2.Y, hue=df2.X);
In [12]:
# Number of outcomes
nYlevels2 = df2.Y.cat.categories.size
# Number of groups
n_grps = df2.X.nunique()
# Group index
grp_idx = df2.X.cat.codes.values
thresh2 = np.arange(1.5, nYlevels2, dtype=np.float32)
thresh_obs2 = np.ma.asarray(thresh2)
thresh_obs2[1:-1] = np.ma.masked
print('thresh2:\t{}'.format(thresh2))
print('thresh_obs2:\t{}'.format(thresh_obs2))
In [13]:
@as_op(itypes=[tt.fvector, tt.fvector, tt.fvector], otypes=[tt.fmatrix])
def outcome_probabilities(theta, mu, sigma):
out = np.empty((nYlevels2, n_grps), dtype=np.float32)
n = norm(loc=mu, scale=sigma)
out[0,:] = n.cdf(theta[0])
out[1,:] = np.max([[0,0], n.cdf(theta[1]) - n.cdf(theta[0])], axis=0)
out[2,:] = np.max([[0,0], n.cdf(theta[2]) - n.cdf(theta[1])], axis=0)
out[3,:] = np.max([[0,0], n.cdf(theta[3]) - n.cdf(theta[2])], axis=0)
out[4,:] = 1 - n.cdf(theta[3])
return out
with pm.Model() as ordinal_model_multi_groups:
theta = pm.Normal('theta', mu=thresh2, tau=np.repeat(.5**2, len(thresh2)),
shape=len(thresh2), observed=thresh_obs2)
mu = pm.Normal('mu', mu=nYlevels2/2.0, tau=1.0/(nYlevels2**2), shape=n_grps)
sigma = pm.Uniform('sigma', nYlevels2/1000.0, nYlevels2*10.0, shape=n_grps)
pr = outcome_probabilities(theta, mu, sigma)
y = pm.Categorical('y', pr[:,grp_idx].T, observed=df2.Y.cat.codes.as_matrix())
pm.model_to_graphviz(ordinal_model_multi_groups)
Out[13]:
In [14]:
with ordinal_model_multi_groups:
trace2 = pm.sample(3000, cores=4)
In [15]:
pm.traceplot(trace2);
In [16]:
mu2 = trace2['mu']
sigma2 = trace2['sigma']
# Concatenate the fixed thresholds into the estimated thresholds
n = trace2['theta_missing'].shape[0]
thresholds2 = np.c_[np.tile([1.5], (n,1)),
trace2['theta_missing'],
np.tile([4.5], (n,1))]
fig, axes = plt.subplots(5,2, figsize=(10,14))
ax1,ax2,ax3,ax4,ax5,ax6,ax7,ax8,ax9,ax10 = axes.flatten()
# Mu
pm.plot_posterior(mu2[:,0], point_estimate='mode', color=color, ax=ax1)
ax1.set_xlabel('$\mu_{1}$', fontdict=f_dict)
pm.plot_posterior(mu2[:,1], point_estimate='mode', color=color, ax=ax3)
ax3.set_xlabel('$\mu_{2}$', fontdict=f_dict)
for title, ax in zip(['A Mean', 'B Mean'], [ax1, ax3]):
ax.set_title(title, fontdict=f_dict)
# Sigma
pm.plot_posterior(sigma2[:,0], point_estimate='mode', color=color, ax=ax5)
ax5.set_xlabel('$\sigma_{1}$', fontdict=f_dict)
pm.plot_posterior(sigma2[:,1], point_estimate='mode', color=color, ax=ax7)
ax7.set_xlabel('$\sigma_{2}$', fontdict=f_dict)
for title, ax in zip(['A Std. Dev.', 'B Std. Dev.'], [ax5, ax7]):
ax.set_title(title, fontdict=f_dict)
# Posterior distribution on the thresholds
ax9.scatter(thresholds2, np.tile(thresholds2.mean(axis=1).reshape(-1,1), (1,4)), color=color, alpha=.6, facecolor='none')
ax9.set_ylabel('Mean Threshold', fontdict=f_dict)
ax9.set_xlabel('Threshold', fontdict=f_dict)
ax9.vlines(x = thresholds2.mean(axis=0),
ymin=thresholds2.mean(axis=1).min(),
ymax=thresholds2.mean(axis=1).max(), linestyles='dotted', colors=color)
# Posterior predictive probabilities of the outcomes
threshCumProb2A = np.empty(thresholds2.shape)
for i in np.arange(threshCumProb2A.shape[0]):
threshCumProb2A[i] = norm().cdf((thresholds2[i] - mu2[i,0])/sigma2[i,0])
outProb2A = (np.c_[threshCumProb2A, np.tile(1, (thresholds2.shape[0],1))]
- np.c_[np.tile(0, (thresholds2.shape[0],1)), threshCumProb2A])
yerr2A = np.abs(np.subtract(pm.hpd(outProb2A), outProb2A.mean(axis=0).reshape(-1,1)))
ax2.errorbar(x = np.arange(outProb2A.shape[1]), y=outProb2A.mean(axis=0),
yerr=yerr2A.T, color=color, fmt='o')
threshCumProb2B = np.empty(thresholds2.shape)
for i in np.arange(threshCumProb2B.shape[0]):
threshCumProb2B[i] = norm().cdf((thresholds2[i] - mu2[i,1])/sigma2[i,1])
outProb2B = (np.c_[threshCumProb2B, np.tile(1, (thresholds2.shape[0],1))]
- np.c_[np.tile(0, (thresholds2.shape[0],1)), threshCumProb2B])
yerr2B = np.abs(np.subtract(pm.hpd(outProb2B), outProb2B.mean(axis=0).reshape(-1,1)))
ax4.errorbar(x = np.arange(outProb2B.shape[1]), y=outProb2B.mean(axis=0),
yerr=yerr2B.T, color=color, fmt='o')
for grp, ax in zip(['A', 'B'], [ax2, ax4]):
((df2[df2.X == grp].Y.value_counts()/df2[df2.X == grp].Y.size)
.plot.bar(ax=ax, rot=0, color='royalblue'))
ax.set_title('Data for {0} with Post. Pred.\nN = {1}'.format(grp, df2[df2.X == grp].Y.size), fontdict=f_dict)
ax.set_xlabel('y')
sns.despine(ax=ax, left=True)
ax.yaxis.set_visible(False)
# Mu diff
pm.plot_posterior(mu2[:,1]-mu2[:,0], point_estimate='mode', color=color, ax=ax6)
ax6.set_xlabel('$\mu_{2}-\mu_{1}$', fontdict=f_dict)
# Sigma diff
pm.plot_posterior(sigma2[:,1]-sigma2[:,0], point_estimate='mode', color=color, ax=ax8)
ax8.set_xlabel('$\sigma_{2}-\sigma_{1}$', fontdict=f_dict)
# Effect size
pm.plot_posterior((mu2[:,1]-mu2[:,0]) / np.sqrt((sigma2[:,0]**2+sigma2[:,1]**2)/2), point_estimate='mode', color=color, ax=ax10)
ax10.set_xlabel(r'$\frac{(\mu_2-\mu_1)}{\sqrt{(\sigma_1^2+\sigma_2^2)/2}}$', fontdict=f_dict)
for title, ax in zip(['Differences of Means', 'Difference of Std. Dev\'s', 'Effect Size'], [ax6, ax8, ax10]):
ax.set_title(title, fontdict=f_dict)
fig.tight_layout()
In [17]:
df3 = pd.read_csv('data/OrdinalProbitData-LinReg-2.csv', dtype={'Y':'category'})
df3.info()
In [18]:
df3.head()
Out[18]:
In [19]:
sd_X = df3.X.std()
mean_X = df3.X.mean()
zX = (df3.X - mean_X)/sd_X
In [20]:
nYlevels3 = df3.Y.nunique()
thresh3 = np.arange(1.5, nYlevels3, dtype=np.float32)
thresh_obs3 = np.ma.asarray(thresh3)
thresh_obs3[1:-1] = np.ma.masked
print('thresh3:\t{}'.format(thresh3))
print('thresh_obs3:\t{}'.format(thresh_obs3))
In [21]:
@as_op(itypes=[tt.fvector, tt.fvector, tt.fscalar], otypes=[tt.fmatrix])
def outcome_probabilities(theta, mu, sigma):
out = np.empty((mu.size, nYlevels3), dtype=np.float32)
n = norm(loc=mu, scale=sigma)
out[:,0] = n.cdf(theta[0])
out[:,1] = np.max([np.repeat(0,mu.size), n.cdf(theta[1]) - n.cdf(theta[0])], axis=0)
out[:,2] = np.max([np.repeat(0,mu.size), n.cdf(theta[2]) - n.cdf(theta[1])], axis=0)
out[:,3] = np.max([np.repeat(0,mu.size), n.cdf(theta[3]) - n.cdf(theta[2])], axis=0)
out[:,4] = np.max([np.repeat(0,mu.size), n.cdf(theta[4]) - n.cdf(theta[3])], axis=0)
out[:,5] = np.max([np.repeat(0,mu.size), n.cdf(theta[5]) - n.cdf(theta[4])], axis=0)
out[:,6] = 1 - n.cdf(theta[5])
return out
with pm.Model() as ordinal_model_metric:
theta = pm.Normal('theta', mu=thresh3, tau=np.repeat(1/2**2, len(thresh3)),
shape=len(thresh3), observed=thresh_obs3)
zbeta0 = pm.Normal('zbeta0', mu=(1+nYlevels3)/2, tau=1/nYlevels3**2)
zbeta = pm.Normal('zbeta', mu=0.0, tau=1/nYlevels3**2)
mu = pm.Deterministic('mu', zbeta0 + zbeta*zX.astype('float32'))
zsigma = pm.Uniform('zsigma', nYlevels3/1000.0, nYlevels3*10.0)
pr = outcome_probabilities(theta, mu, zsigma)
y = pm.Categorical('y', pr, observed=df3.Y.cat.codes)
pm.model_to_graphviz(ordinal_model_metric)
Out[21]:
In [22]:
with ordinal_model_metric:
trace3 = pm.sample(3000, cores=4)
In [23]:
pm.traceplot(trace3);
In [24]:
# Convert parameters to original scale
beta = trace3['zbeta']/sd_X
beta0 = trace3['zbeta0'] - trace3['zbeta']*mean_X/sd_X
sigma = trace3['zsigma']
# Concatenate the fixed thresholds into the estimated thresholds
n = trace3['theta_missing'].shape[0]
thresholds3 = np.c_[np.tile([1.5], (n,1)),
trace3['theta_missing'],
np.tile([6.5], (n,1))]
# Define gridspec
fig = plt.figure(figsize=(10,10))
gs = gridspec.GridSpec(4, 3)
ax1 = plt.subplot(gs[:2,:])
ax2 = plt.subplot(gs[2,0])
ax3 = plt.subplot(gs[2,1])
ax4 = plt.subplot(gs[2,2])
ax5 = plt.subplot(gs[3,:])
# Scatterplot
ax1.scatter(df3.X, df3.Y, edgecolors='k', lw=2, facecolor='none')
# Samples of regression lines
x_range = np.linspace(df3.X.min(), df3.X.max())
B = pd.DataFrame(np.c_[beta0, beta], columns=['beta0', 'beta']).sample(20)
for i in np.arange(len(B)):
ax1.plot(x_range, B.iloc[i,0]+B.iloc[i,1]*x_range, c=color, alpha=0.5)
ax1.set_ylim((0.5,7.75))
ax1.set_xlim(xmin=.8)
# Draw the posterior (mean) predicted probability at 5 selected values of the predictor.
# Not stepping through the chain in order to calculate the HDI.
for v in np.linspace(df3.X.min(), df3.X.max(), 5):
ax1.axvline(x=v, color='grey', alpha=.5)
mu = beta0.mean()+beta.mean()*v
threshCumProb3 = norm().cdf((np.mean(thresholds3, axis=0) - mu)/sigma.mean())
outProb3 = np.diff(np.r_[0, threshCumProb3, 1])
for i, p in enumerate(outProb3):
ax1.add_patch(Rectangle(xy=(v-p/10, i+0.75), width=p/10, height=0.75, color=color, alpha=.5))
pm.plot_posterior(beta0, point_estimate='mode', color=color, ax=ax2)
pm.plot_posterior(beta, point_estimate='mode', color=color, ax=ax3)
pm.plot_posterior(sigma, point_estimate='mode', color=color, ax=ax4);
for title, label, ax in zip(['Intercept', 'X', 'Std. Dev.'],
[r'$\beta_{0}$', r'$\beta_{1}$', r'$\sigma$'],
[ax2, ax3, ax4]):
ax.set_title(title, fontdict=f_dict)
ax.set_xlabel(label, fontdict=f_dict)
# Posterior distribution on the thresholds
ax5.scatter(thresholds3, np.tile(thresholds3.mean(axis=1).reshape(-1,1), (1,6)), color=color, alpha=.6, facecolor='none')
ax5.set_ylabel('Mean Threshold', fontdict=f_dict)
ax5.set_xlabel('Threshold', fontdict=f_dict)
ax5.vlines(x = thresholds3.mean(axis=0),
ymin=thresholds3.mean(axis=1).min(),
ymax=thresholds3.mean(axis=1).max(), linestyles='dotted', colors=color)
fig.tight_layout()
In [25]:
df4 = pd.read_csv('data/Movies.csv', usecols=[1,2,4], dtype={'Rating':'category'})
df4.info()
In [26]:
df4.head()
Out[26]:
In [27]:
X = df4[['Year','Length']]
Y = df4.Rating
In [28]:
sd_X = X.std()
mean_X = X.mean()
zX = (X - mean_X)/sd_X
In [29]:
nYlevels4 = Y.nunique()
thresh4 = np.arange(1.5, nYlevels4, dtype=np.float32)
thresh_obs4 = np.ma.asarray(thresh4)
thresh_obs4[1:-1] = np.ma.masked
print('thresh4:\t{}'.format(thresh4))
print('thresh_obs4:\t{}'.format(thresh_obs4))
In [30]:
@as_op(itypes=[tt.fvector, tt.fvector, tt.fscalar], otypes=[tt.fmatrix])
def outcome_probabilities(theta, mu, sigma):
out = np.empty((mu.size, nYlevels4), dtype=np.float32)
n = norm(loc=mu, scale=sigma)
out[:,0] = n.cdf(theta[0])
out[:,1] = np.max([np.repeat(0,mu.size), n.cdf(theta[1]) - n.cdf(theta[0])], axis=0)
out[:,2] = np.max([np.repeat(0,mu.size), n.cdf(theta[2]) - n.cdf(theta[1])], axis=0)
out[:,3] = np.max([np.repeat(0,mu.size), n.cdf(theta[3]) - n.cdf(theta[2])], axis=0)
out[:,4] = np.max([np.repeat(0,mu.size), n.cdf(theta[4]) - n.cdf(theta[3])], axis=0)
out[:,5] = np.max([np.repeat(0,mu.size), n.cdf(theta[5]) - n.cdf(theta[4])], axis=0)
out[:,6] = 1 - n.cdf(theta[5])
return out
with pm.Model() as ordinal_model_multi_metric:
theta = pm.Normal('theta', mu=thresh4, tau=np.repeat(1/2**2, len(thresh4)),
shape=len(thresh4), observed=thresh_obs4)
zbeta0 = pm.Normal('zbeta0', mu=(1+nYlevels4)/2, tau=1/nYlevels4**2)
zbeta = pm.Normal('zbeta', mu=0.0, tau=1/nYlevels4**2, shape=X.shape[1])
mu = pm.Deterministic('mu', zbeta0 + pm.math.dot(zbeta,zX.T.astype('float32')))
zsigma = pm.Uniform('zsigma', nYlevels4/1000.0, nYlevels4*10.0)
pr = outcome_probabilities(theta, mu, zsigma)
y = pm.Categorical('y', pr, observed=Y.cat.codes)
pm.model_to_graphviz(ordinal_model_multi_metric)
Out[30]:
In [34]:
with ordinal_model_multi_metric:
trace4 = pm.sample(3000, cores=4)
In [35]:
pm.traceplot(trace4);
In [36]:
# Convert parameters to original scale
beta = trace4['zbeta']/sd_X.values
beta0 = trace4['zbeta0'] - np.sum(trace4['zbeta']*mean_X.values/sd_X.values, axis=1)
sigma = trace4['zsigma']
# Concatenate the fixed thresholds into the estimated thresholds
n = trace4['theta_missing'].shape[0]
thresholds4 = np.c_[np.tile([1.5], (n,1)),
trace4['theta_missing'],
np.tile([6.5], (n,1))]
# Define gridspec
fig = plt.figure(figsize=(10,14))
gs = gridspec.GridSpec(6, 3)
ax1 = plt.subplot(gs[:4,:])
ax2 = plt.subplot(gs[4,0])
ax3 = plt.subplot(gs[4,1])
ax4 = plt.subplot(gs[4,2])
ax5 = plt.subplot(gs[5,:])
for year, length, marker in zip(df4.Year, df4.Length, df4.Rating.cat.codes.map(lambda m: r'${}$'.format(m))):
ax1.scatter(year, length, marker=marker, s=100, c='k')
ax1.set_xlabel('Year', fontdict=f_dict)
ax1.set_ylabel('Length', fontdict=f_dict)
ax1.set_xlim((df4.Year.min()-5,df4.Year.max()+5))
ax1.set_ylim((df4.Length.min()*.95,df4.Length.max()*1.05))
# Plot three sets of thresholds
# Randomly selecting 3 steps from the trace
sample_size = 3
trace_idx = np.random.randint(0,high=len(trace4), size=sample_size)
# Different colors for each of the 3 steps
line_colors = ['red', 'green', 'blue']
x1_year = np.linspace(df4.Year.min()-5, df4.Year.max()+5)
# Looping over the three sample indexes and six thresholds simultaneously (3x6 matrix)
for i, k in np.ndindex(sample_size,thresholds4.shape[1]):
idx = trace_idx[i]
# Equation 23.5
x2_length = (thresholds4[idx,k]-beta0[idx])/beta[idx,1]+(-beta[idx,0]/beta[idx,1])*x1_year
ax1.plot(x1_year, x2_length, c=line_colors[i])
# Plot posteriors
pm.plot_posterior(beta0, point_estimate='mode', color=color, ax=ax2)
pm.plot_posterior(beta[:,0], point_estimate='mode', color=color, ax=ax3)
pm.plot_posterior(beta[:,1], point_estimate='mode', color=color, ax=ax4);
for title, label, ax in zip(['Intercept', 'Year', 'Length'],
[r'$\beta_{0}$', r'$\beta_{1}$', r'$\beta_{2}$'],
[ax2, ax3, ax4]):
ax.set_title(title, fontdict=f_dict)
ax.set_xlabel(label, fontdict=f_dict)
# Posterior distribution on the thresholds
ax5.scatter(thresholds4, np.tile(thresholds4.mean(axis=1).reshape(-1,1), (1,6)), color=color, alpha=.6, facecolor='none')
ax5.set_ylabel('Mean Threshold', fontdict=f_dict)
ax5.set_xlabel('Threshold', fontdict=f_dict)
ax5.vlines(x = thresholds4.mean(axis=0),
ymin=thresholds4.mean(axis=1).min(),
ymax=thresholds4.mean(axis=1).max(), linestyles='dotted', colors=color)
fig.tight_layout()