# We can see from this plot that in order to take a representative sample,
# some early portion of the chain must be removed. This is referred to as
# the 'burn-in' period. This period allows the chain to both find the high
# density areas, and adjust the proposal widths to their optimal values.
# The plot_diagnostics() method can help us decide what size of burn-in to use:
chain.plot_diagnostics()
# Occasionally samples are also 'thinned' by a factor of n (where only every
# n'th sample is used) in order to reduce the size of the data set for
# storage, or to produce uncorrelated samples.
# based on the diagnostics we can choose to manually set a global burn and
# thin value, which is used (unless otherwise specified) by all methods which
# access the samples
chain.burn = 2000
chain.thin = 5
# the burn-in and thinning can also be set automatically as follows:
chain.autoselect_burn_and_thin()
# After discarding burn-in, what we have left should be a representative
# sample drawn from the posterior. Repeating the previous plot as a
# scatter-plot shows the sample:
p = chain.get_probabilities() # color the points by their probability value
plt.scatter(chain.get_parameter(0),
chain.get_parameter(1),
c=exp(p - max(p)),
marker='.')
plt.xlabel('parameter 1')
plt.ylabel('parameter 2')
# We can see from this plot that in order to take a representative sample,
# some early portion of the chain must be removed. This is referred to as
# the 'burn-in' period. This period allows the chain to both find the high
# density areas, and adjust the proposal widths to their optimal values.
# The plot_diagnostics() method can help us decide what size of burn-in to use:
chain.plot_diagnostics(filename='gibbs_diagnostics.png')
# Occasionally samples are also 'thinned' by a factor of n (where only every
# n'th sample is used) in order to reduce the size of the data set for
# storage, or to produce uncorrelated samples.
# based on the diagnostics we can choose to manually set a global burn and
# thin value, which is used (unless otherwise specified) by all methods which
# access the samples
chain.burn = 10000
chain.thin = 10
# the burn-in and thinning can also be set automatically as follows:
# chain.autoselect_burn_and_thin()
# After discarding burn-in, what we have left should be a representative
# sample drawn from the posterior. Repeating the previous plot as a
# scatter-plot shows the sample:
p = chain.get_probabilities() # color the points by their probability value
plt.scatter(chain.get_parameter(0),
chain.get_parameter(1),
c=exp(p - max(p)),
marker='.')
plt.xlabel('parameter 1')
plt.ylabel('parameter 2')
# plot the synthetic data and underlying line plt.plot(x, m*x + c) plt.plot(x, y, '.') plt.grid() plt.show() # create an instance of the posterior class posterior = LinePosterior(x = x, y = y, err = ones(N)*sigma) # pass the posterior to the MCMC sampler chain = GibbsChain(posterior = posterior, start = [0.5, 0.1]) # Now suppose we know the offset parameter must be non-negative. # This constraint can be imposed by passing the index of the # parameter to the set_non_negative method as follows: chain.set_non_negative(1) # For the purposes of this demo, let's assume we also know that # the gradient must exist in the range [0.45, 0.55]. # The gradient can be constrained to values between chosen boundaries # by passing the parameter index and the boundary values to the # set_boundaries method as follows: chain.set_boundaries(0, [0.45, 0.55]) # Advance the chain chain.advance(50000) chain.burn = 5000 # Use the matrix plot functionality to check the constraints are working chain.matrix_plot()
import matplotlib.pyplot as plt
def rosenbrock(t):
x, y = t
x2 = x**2
b = 15. # correlation strength parameter
v = 3. # variance of the gaussian term
return -x2 - b * (y - x2)**2 - 0.5 * (x2 + y**2) / v
# create the chain object
from inference.mcmc import GibbsChain
gibbs = GibbsChain(posterior=rosenbrock, start=array([2., -4.]))
gibbs.advance(150000)
gibbs.burn = 10000
gibbs.thin = 70
p = gibbs.get_probabilities() # color the points by their probability value
fig = plt.figure(figsize=(5, 4))
ax1 = fig.add_subplot(111)
ax1.scatter(gibbs.get_parameter(0),
gibbs.get_parameter(1),
c=exp(p - max(p)),
marker='.')
ax1.set_ylim([None, 2.8])
ax1.set_xlim([-1.8, 1.8])
ax1.set_xticks([])
ax1.set_yticks([])
# ax1.set_title('Gibbs sampling')
plt.tight_layout()
