def plot_density():
omegas = np.linspace(1e-16, 5, 1000)
logpomega = log_polya_gamma_density(omegas, b, 0, trunc=1000)
pomega = np.exp(logpomega).real
import matplotlib.pyplot as plt
plt.ion()
plt.figure()
plt.plot(omegas, pomega)
y = -b * np.log(2) + (a-b/2.) * psi -0.5*omegas*psi**2
from scipy.integrate import simps
Z = simps(pomega, omegas)
print("Z: ", Z)
def simps_approx():
# Compute the left hand side analytically
loglhs = psi*a - b * np.log1p(np.exp(psi))
lhs = np.exp(loglhs)
# Compute the right hand side with quadrature
from scipy.integrate import simps
# Lay down a grid of omegas
# TODO: How should we choose the bins?
omegas = np.linspace(1e-15, 5, 1000)
# integrand = lambda om: 2**-b \
# * np.exp((a-b/2.)*psi 0 0.5*om*psi**2) \
# * polya_gamma_density(om, b, 0)
# y = map(integrand, omegas)
# rhs = simps(integrand, y)
logy = -b * np.log(2) + (a-b/2.) * psi -0.5*omegas*psi**2
logy += log_polya_gamma_density(omegas, b, 0, trunc=21)
y = np.exp(logy)
rhs = simps(y, omegas)
print("Numerical Quadrature")
print("log LHS: ", loglhs)
print("log RHS: ", np.log(rhs))