Eggbox

A likelihood surface with multiple modes of equal height.

import numpy as np
import matplotlib.pyplot as plt
import corner

import nestle


# Define the posterior density to be sampled:
tmax = 5.0 * np.pi
constant = np.log(1.0 / tmax**2)

def loglike(x):
    t = 2.0 * tmax * x - tmax
    return (2.0 + np.cos(t[0]/2.0)*np.cos(t[1]/2.0))**5.0

def prior(x):
    return x

# plot the surface
plt.figure(figsize=(8., 8.))
ax = plt.axes(aspect=1)
xx, yy = np.meshgrid(np.linspace(0., 1., 50),
                     np.linspace(0., 1., 50))
Z = loglike(np.array([xx, yy]))
ax.contourf(xx, yy, Z, 12, cmap=plt.cm.Blues_r)
plt.title("True Log likelihood surface")

Text(0.5, 1.0, 'True Log likelihood surface')

Run nested sampling in multi-ellipsoid mode and print a summary of results:

res = nestle.sample(loglike, prior, 2, npoints=200, method='multi',
                    update_interval=20)
print(res.summary())

niter: 1532
ncall: 3095
nsamples: 1732
logz: 236.196 +/-  0.170
h:  5.806

Plot the samples. Note that this represents the likelihood rather than its log, hence it is much more highly peaked.

fig = corner.corner(res.samples, weights=res.weights, bins=500,
                    range=[(0., 1.), (0., 1.)])
fig.set_size_inches(8., 8.)