def testDensityIncreasingStepAccepted(self):
"""Tests that if a transition increases density, it is always accepted."""
target_log_density = lambda x: -x * x
state = variable_scope.get_variable("state", initializer=10.)
state_log_density = variable_scope.get_variable(
"state_log_density",
initializer=target_log_density(state.initialized_value()))
log_accept_ratio = variable_scope.get_variable("log_accept_ratio",
initializer=0.)
get_next_proposal = lambda x: (x - 1., None)
step = mh.evolve(state,
state_log_density,
log_accept_ratio,
target_log_density,
get_next_proposal,
seed=1234)
init = variables.initialize_all_variables()
with self.test_session() as sess:
sess.run(init)
for j in range(9):
sess.run(step)
sample = sess.run(state)
sample_log_density = sess.run(state_log_density)
self.assertAlmostEqual(sample, 9 - j)
self.assertAlmostEqual(sample_log_density, -(9 - j) * (9 - j))
def testDocstringExample(self):
"""Tests the simplified docstring example with multiple chains."""
n = 2 # dimension of the problem
# Generate 300 initial values randomly. Each of these would be an
# independent starting point for a Markov chain.
state = variable_scope.get_variable(
"state", initializer=random_ops.random_normal(
[300, n], mean=3.0, dtype=dtypes.float32, seed=42))
# Computes the log(p(x)) for the unit normal density and ignores the
# normalization constant.
def log_density(x):
return - math_ops.reduce_sum(x * x, reduction_indices=-1) / 2.0
# Initial log-density value
state_log_density = variable_scope.get_variable(
"state_log_density",
initializer=log_density(state.initialized_value()))
# A variable to store the log_acceptance_ratio:
log_acceptance_ratio = variable_scope.get_variable(
"log_acceptance_ratio",
initializer=array_ops.zeros([300], dtype=dtypes.float32))
# Generates random proposals by moving each coordinate uniformly and
# independently in a box of size 2 centered around the current value.
# Returns the new point and also the log of the Hastings ratio (the
# ratio of the probability of going from the proposal to origin and the
# probability of the reverse transition). When this ratio is 1, the value
# may be omitted and replaced by None.
def random_proposal(x):
return (x + random_ops.random_uniform(
array_ops.shape(x), minval=-1, maxval=1,
dtype=x.dtype, seed=12)), None
# Create the op to propagate the chain for 100 steps.
stepper = mh.evolve(
state, state_log_density, log_acceptance_ratio,
log_density, random_proposal, n_steps=100, seed=123)
init = variables.initialize_all_variables()
with self.test_session() as sess:
sess.run(init)
# Run the chains for a total of 1000 steps.
for _ in range(10):
sess.run(stepper)
samples = sess.run(state)
covariance = np.eye(n)
# Verify that the estimated mean and covariance are close to the true
# values.
self.assertAlmostEqual(
np.max(np.abs(np.mean(samples, 0)
- np.zeros(n))), 0,
delta=0.1)
self.assertAlmostEqual(
np.max(np.abs(np.reshape(np.cov(samples, rowvar=False), [n**2])
- np.reshape(covariance, [n**2]))), 0,
delta=0.2)
def testSampleProperties(self):
"""Tests that the samples converge to the target distribution."""
def target_log_density(x):
"""Log-density corresponding to a normal distribution with mean = 4."""
return -(x - 2.0) * (x - 2.0) * 0.5
# Use the uniform random walker to generate proposals.
proposal_fn = mh.proposal_uniform(step_size=1.0, seed=1234)
state = variable_scope.get_variable("state", initializer=0.0)
state_log_density = variable_scope.get_variable(
"state_log_density",
initializer=target_log_density(state.initialized_value()))
log_accept_ratio = variable_scope.get_variable("log_accept_ratio",
initializer=0.)
# Random walk MCMC converges slowly so need to put in enough iterations.
num_iterations = 5000
step = mh.evolve(state,
state_log_density,
log_accept_ratio,
target_log_density,
proposal_fn,
seed=4321)
init = variables.global_variables_initializer()
sample_sum, sample_sq_sum = 0.0, 0.0
with self.test_session() as sess:
sess.run(init)
for _ in np.arange(num_iterations):
# Allow for the mixing of the chain and discard these samples.
sess.run(step)
for _ in np.arange(num_iterations):
sess.run(step)
sample = sess.run(state)
sample_sum += sample
sample_sq_sum += sample * sample
sample_mean = sample_sum / num_iterations
sample_variance = sample_sq_sum / num_iterations - sample_mean * sample_mean
# The samples have large autocorrelation which reduces the effective sample
# size.
self.assertAlmostEqual(sample_mean, 2.0, delta=0.1)
self.assertAlmostEqual(sample_variance, 1.0, delta=0.1)
def testSampleProperties(self):
"""Tests that the samples converge to the target distribution."""
def target_log_density(x):
"""Log-density corresponding to a normal distribution with mean = 4."""
return - (x - 2.0) * (x - 2.0) * 0.5
# Use the uniform random walker to generate proposals.
proposal_fn = mh.proposal_uniform(
step_size=1.0, seed=1234)
state = variable_scope.get_variable("state", initializer=0.0)
state_log_density = variable_scope.get_variable(
"state_log_density",
initializer=target_log_density(state.initialized_value()))
log_accept_ratio = variable_scope.get_variable(
"log_accept_ratio", initializer=0.)
# Random walk MCMC converges slowly so need to put in enough iterations.
num_iterations = 5000
step = mh.evolve(state, state_log_density, log_accept_ratio,
target_log_density, proposal_fn, seed=4321)
init = variables.global_variables_initializer()
sample_sum, sample_sq_sum = 0.0, 0.0
with self.test_session() as sess:
sess.run(init)
for _ in np.arange(num_iterations):
# Allow for the mixing of the chain and discard these samples.
sess.run(step)
for _ in np.arange(num_iterations):
sess.run(step)
sample = sess.run(state)
sample_sum += sample
sample_sq_sum += sample * sample
sample_mean = sample_sum / num_iterations
sample_variance = sample_sq_sum / num_iterations - sample_mean * sample_mean
# The samples have large autocorrelation which reduces the effective sample
# size.
self.assertAlmostEqual(sample_mean, 2.0, delta=0.1)
self.assertAlmostEqual(sample_variance, 1.0, delta=0.1)
def testDensityIncreasingStepAccepted(self):
"""Tests that if a transition increases density, it is always accepted."""
target_log_density = lambda x: - x * x
state = variable_scope.get_variable("state", initializer=10.)
state_log_density = variable_scope.get_variable(
"state_log_density",
initializer=target_log_density(state.initialized_value()))
log_accept_ratio = variable_scope.get_variable(
"log_accept_ratio", initializer=0.)
get_next_proposal = lambda x: (x - 1., None)
step = mh.evolve(state, state_log_density, log_accept_ratio,
target_log_density, get_next_proposal, seed=1234)
init = variables.initialize_all_variables()
with self.test_session() as sess:
sess.run(init)
for j in range(9):
sess.run(step)
sample = sess.run(state)
sample_log_density = sess.run(state_log_density)
self.assertAlmostEqual(sample, 9 - j)
self.assertAlmostEqual(sample_log_density, - (9 - j) * (9 - j))
def testDocstringExample(self):
"""Tests the simplified docstring example with multiple chains."""
n = 2 # dimension of the problem
# Generate 300 initial values randomly. Each of these would be an
# independent starting point for a Markov chain.
state = variable_scope.get_variable(
"state",
initializer=random_ops.random_normal([300, n],
mean=3.0,
dtype=dtypes.float32,
seed=42))
# Computes the log(p(x)) for the unit normal density and ignores the
# normalization constant.
def log_density(x):
return -math_ops.reduce_sum(x * x, reduction_indices=-1) / 2.0
# Initial log-density value
state_log_density = variable_scope.get_variable(
"state_log_density",
initializer=log_density(state.initialized_value()))
# A variable to store the log_acceptance_ratio:
log_acceptance_ratio = variable_scope.get_variable(
"log_acceptance_ratio",
initializer=array_ops.zeros([300], dtype=dtypes.float32))
# Generates random proposals by moving each coordinate uniformly and
# independently in a box of size 2 centered around the current value.
# Returns the new point and also the log of the Hastings ratio (the
# ratio of the probability of going from the proposal to origin and the
# probability of the reverse transition). When this ratio is 1, the value
# may be omitted and replaced by None.
def random_proposal(x):
return (x + random_ops.random_uniform(array_ops.shape(x),
minval=-1,
maxval=1,
dtype=x.dtype,
seed=12)), None
# Create the op to propagate the chain for 100 steps.
stepper = mh.evolve(state,
state_log_density,
log_acceptance_ratio,
log_density,
random_proposal,
n_steps=100,
seed=123)
init = variables.initialize_all_variables()
with self.test_session() as sess:
sess.run(init)
# Run the chains for a total of 1000 steps.
for _ in range(10):
sess.run(stepper)
samples = sess.run(state)
covariance = np.eye(n)
# Verify that the estimated mean and covariance are close to the true
# values.
self.assertAlmostEqual(np.max(
np.abs(np.mean(samples, 0) - np.zeros(n))),
0,
delta=0.1)
self.assertAlmostEqual(np.max(
np.abs(
np.reshape(np.cov(samples, rowvar=False), [n**2]) -
np.reshape(covariance, [n**2]))),
0,
delta=0.2)