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2d_post_comparison.py
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import matplotlib.pyplot as plt
from matplotlib import rc
import numpy as np
from scipy.stats import multivariate_normal
np.random.seed(104)
# ===============================
# this code creates figure 3 plot
# from
# Uncertainty in Neural Networks:
# Approximately Bayesian Ensembling
# ===============================
# inputs
n_samples = 50
is_save = 0
type_post = 'general' # general, correlated, extrapolation
# prior
prior_mean = [0.,0.]
lambda_1 = 1. # from eq. 31
prior_cov = lambda_1*np.array([[1.,0.],[0.,1]])
prior_cov_inv = np.linalg.inv(prior_cov)
for type_inf in ['analytical','anchor','vi']:
# likelihood
#
if type_post == 'general':
like_mean = [1.,-1]
corr = 0.8
a = 1.*1.5
d = 1.*1.5
elif type_post == 'extrapolation':
like_mean = [1.*0.5,-1*0.5] # use for vi plots
corr = 0.
a = 1.*2000
d = 1.*2000
elif type_post == 'correlated':
like_mean = [1.*0.5,-1*0.5] # use for vi plots
corr = 0.9999
a = 1.*2
d = 1.*2
else:
raise Exception
cov_off_diag = corr*np.sqrt(a*d)
like_cov = np.array([[a,cov_off_diag],[cov_off_diag,d]]) # use for vi
like_cov_inv = np.linalg.inv(like_cov)
# set up plot
rc('legend',**{'fontsize':19})
plt.rcParams['pdf.fonttype'] = 42
plt.rcParams['ps.fonttype'] = 42
plt.rcParams.update({'font.size': 32})
plt.rcParams['text.usetex'] = True
xlimits = [-2.1, 3.8]
ylimits = [-2.7, 2.7]
x = np.linspace(*xlimits, num=2000)
y = np.linspace(*ylimits, num=2000)
X, Y = np.meshgrid(x, y)
fig = plt.figure(figsize=(5, 5)); fig.clf()
ax = fig.add_subplot(111)
# prior
rv = multivariate_normal(prior_mean, prior_cov)
pos = np.empty(X.shape + (2,))
pos[:, :, 0] = X
pos[:, :, 1] = Y
prior = rv.pdf(pos)
CS2 = plt.contour(X, Y, prior, 2, colors='r',alpha=1.0,lw=1)
# CS2.collections[0].set_label('Prior')
# likelihood
rv = multivariate_normal(like_mean, like_cov)
like = rv.pdf(pos)
# CS1 = plt.contour(X, Y, like, 2, colors='k',alpha=1.0, lw=1)
# CS1 = plt.contour(X, Y, like, 2, colors='k',alpha=1.0, linewidths=2., levels = [ 0.09])
# CS1.collections[0].set_label('Likelihood')
# calculate anchor dist
ident = np.identity(2)
# anch_cov_analy = np.matmul(np.matmul((ident + lambda_1*like_cov_inv),np.linalg.inv(( (1/lambda_1)*ident + like_cov_inv))),(ident + lambda_1*like_cov_inv))
# anch_cov_analy = prior_cov + np.matmul(np.matmul(prior_cov,prior_cov) , like_cov_inv)
anch_cov_analy = prior_cov + np.matmul(np.matmul(prior_cov,like_cov_inv),prior_cov)
anch_cov_analy = prior_cov # if want to overwrite and use workaround
# anch_cov_analy = lambda_1*ident + lambda_1**2 * like_cov_inv
anch_cov_analy_inv = np.linalg.inv(anch_cov_analy)
anch_mean = prior_mean.copy()
# disp_name = 'Anchor dist. exact'
disp_name = 'Initialisation dist.'
save_name = 'MAP_recon_exact_2.eps'
# anchor
rv = multivariate_normal(anch_mean, anch_cov_analy)
anchor = rv.pdf(pos)
# analytical posterior
post_cov_analy = np.linalg.inv(prior_cov_inv+like_cov_inv)
post_mean_analy = np.matmul(np.matmul(post_cov_analy,prior_cov_inv),prior_mean)\
+ np.matmul(np.matmul(post_cov_analy,like_cov_inv),like_mean)
rv_post_analy = multivariate_normal(post_mean_analy, post_cov_analy)
# posterior here
if type_inf=='analytical':
if type_post == 'correlated':
CS4 = plt.contour(X, Y, rv_post_analy.pdf(pos)/rv_post_analy.pdf(pos).max(), 2, colors='g',alpha=1.0,linewidths=3.,levels = [0.2])
elif type_post == 'extrapolation':
CS4 = plt.contour(X, Y, rv_post_analy.pdf(pos)/rv_post_analy.pdf(pos).max(), 2, colors='g',alpha=1.0,linewidths=3.,levels = [0.35])
else:
CS4 = plt.contour(X, Y, rv_post_analy.pdf(pos)/rv_post_analy.pdf(pos).max(), 2, colors='g',alpha=1.0,linewidths=3.,levels = [0.35])
CS4.collections[0].set_label('True post.') # analytical
# anchored post
elif type_inf=='anchor':
rv_post_anch_analy = multivariate_normal(post_mean_analy, np.matmul(np.matmul(post_cov_analy,prior_cov_inv),post_cov_analy))
if type_post == 'correlated':
CS4 = plt.contour(X, Y, rv_post_anch_analy.pdf(pos)/rv_post_anch_analy.pdf(pos).max(), 2, colors='purple',alpha=1.0,linewidths=5.,levels = [0.2])
elif type_post == 'extrapolation':
CS4 = plt.contour(X, Y, rv_post_anch_analy.pdf(pos)/rv_post_anch_analy.pdf(pos).max(), 2, colors='purple',alpha=1.0,linewidths=5.,levels = [0.35])
else:
CS4 = plt.contour(X, Y, rv_post_anch_analy.pdf(pos)/rv_post_anch_analy.pdf(pos).max(), 2, colors='purple',alpha=1.0,linewidths=3.,levels = [0.35])
CS4.collections[0].set_label('RMS approx post.') # analytical
elif type_inf=='vi':
# VI post
vi_cov_analy_post = np.eye(post_cov_analy.shape[0])
post_cov_analy_inv = np.linalg.inv(post_cov_analy)
vi_cov_analy_post[0,0] = 1/post_cov_analy_inv[1,1]
vi_cov_analy_post[1,1] = 1/post_cov_analy_inv[0,0]
rv_vi_post_analy = multivariate_normal(post_mean_analy, vi_cov_analy_post)
if type_post == 'correlated':
CS5 = ax.contour(X, Y, rv_vi_post_analy.pdf(pos)/rv_vi_post_analy.pdf(pos).max(), 2, colors='blue',alpha=1.,lw=2,linewidths=3.,levels = [0.2])
elif type_post == 'extrapolation':
CS5 = ax.contour(X, Y, rv_vi_post_analy.pdf(pos)/rv_vi_post_analy.pdf(pos).max(), 2, colors='blue',alpha=1.,lw=2,linewidths=3.,levels = [0.35])
else:
CS5 = ax.contour(X, Y, rv_vi_post_analy.pdf(pos)/rv_vi_post_analy.pdf(pos).max(), 2, colors='blue',alpha=1.,lw=2,linewidths=3.,levels = [0.35])
CS5.collections[0].set_label('MFVI post.')
else:
raise Exception
# centre of prior
ax.scatter(prior_mean[0],prior_mean[1],c='k', marker='+', s=100)
# ax.set_xlabel('Parameter 1')
# ax.set_ylabel('Parameter 2')
# ax.set_xlabel(r'$\theta_1$', fontsize=12)
# ax.set_ylabel(r'$\theta_2$', fontsize=12)
ax.legend(loc='upper right')
ax.set_yticklabels([])
ax.set_xticklabels([])
ax.set_xticks([])
ax.set_yticks([])
# fig.set_size_inches((5,4))
fig.show()
plt.show(block = False)
if is_save:
fig.savefig('general_vi'+ type_inf +'.pdf', format='pdf', dpi=1000, bbox_inches='tight')
plt.show()