def plot_posterior(prior, Nt=100):
"""
Plot the true (analytic) posterior density function along with the likelihood,
prior and true value of theta.
Parameters:
-----------
prior : array_like ([dist_type, param1, param2])
Vector defining the prior distribution through the dist_type ('normal'
or 'unif'), and two parameters (floats, mean and std for Gaussian, LHS
and range for Uniform).
Nt : int
Number of points at which to evaluate each density function.
Returns:
--------
Plot the true (analytic) posterior density function along with the likelihood,
prior and true value of theta.
"""
# Theta values for evaluating the posterior.
tv = np.linspace(0,3,Nt)
# Evaluate posterior, prior and likelihood.
pv, p0v, Lv = posterior(tv, prior, ip_data)
# Normalise.
p0v = p0v/np.trapz(p0v,tv)
Lv = Lv/np.trapz(Lv,tv)
pv = pv/np.trapz(pv,tv)
# Plot.
fig, ax = myfigure()
ax.plot(tv, pv, label='posterior')
ax.plot(tv, Lv, label='likelihood')
ax.plot(tv, p0v, label='prior')
plt.axvline(x=np.log(ip_data['lam']), color='c', label='truth')
ax.set_xlabel('theta')
ax.set_ylabel('probability density')
leg = ax.legend();
plt.show(block=False)
def hist_plot(hist, bin_centres):
"""
Plot a histogram of the input Markov chain.
Parameters:
-----------
hist : array
Probability density values.
bin_centres : array
Bin centre values.
Returns:
--------
Plot the input histogram.
"""
# Plot histogram (manually).
fig, ax = myfigure()
ax.plot(bin_centres, hist, label='posterior')
plt.axvline(x=np.log(ip_data['lam']), color='r', label='truth')
ax.set_xlabel('theta')
ax.set_ylabel('probability density')
leg = ax.legend();
plt.show()
pv = np.zeros((3, Nt))
# Evaluate posterior.
for n in range(Nt):
pv[0, n], p0v[0, n], dummy1 = util.posterior(tv[n], prior1, ip_data)
pv[1, n], p0v[1, n], dummy2 = util.posterior(tv[n], prior2, ip_data)
pv[2, n], p0v[2, n], dummy3 = util.posterior(tv[n], prior3, ip_data)
# Normalise.
for i in range(3):
p0v[i] = p0v[i] / np.trapz(p0v[i], tv)
pv[i] = pv[i] / np.trapz(pv[i], tv)
## PLOT
fig, ax = myfigure()
ax.plot(tv, pv[0], '-b', label='posterior1')
ax.plot(tv, p0v[0], ':b', label='prior1')
ax.plot(tv, pv[1], '-r', label='posterior2')
ax.plot(tv, p0v[1], ':r', label='prior2')
ax.plot(tv, pv[2], '-g', label='posterior3')
ax.plot(tv, p0v[2], ':g', label='prior3')
plt.axvline(x=np.log(ip_data['lam']), color='c', label='truth')
ax.set_xlabel('theta')
ax.set_ylabel('probability density')
leg = ax.legend()
plt.show()
## ----------
## PLOT DATA:
## ----------
# Spatial locations (plotting).
x = np.linspace(0,1,50)
# Matrix representation of noisy data.
d_mat = np.reshape(ip_data['d'],(ip_data['nt'],ip_data['nx']))
# Labels for y-axes.
ylabs = ['u(x,0)', 'u(x,T/3)','u(x,2T/3)','u(x,T)']
# Initiate 2x2 plot.
fig, ax = myfigure(nrows=2, ncols=2)
for i,axi in enumerate(ax):
if i == 0:
# Initial data.
axi.plot(x, np.squeeze(uxt(x,0,ip_data['lam'])))
else:
# Measurement times data.
axi.plot(x, np.squeeze(uxt(x,ip_data['t_obs'][i-1],ip_data['lam'])))
axi.plot(ip_data['x_obs'], d_mat[i-1,:], 'xr')
axi.set_xlabel('x')
axi.set_ylabel(ylabs[i])
axi.axis([0, 1, 0, 3])