from collections import namedtuple
import jax
from jax import device_get, grad, lax, random, value_and_grad, vmap
from jax.flatten_util import ravel_pytree
import jax.numpy as np
from jax.ops import index_update
from jax.scipy.special import expit
from jax.tree_util import tree_flatten, tree_map, tree_multimap
import numpyro.distributions as dist
from numpyro.distributions.constraints import ComposeTransform, biject_to, real
from numpyro.distributions.util import cholesky_inverse, get_dtype
from numpyro.handlers import seed, trace
from numpyro.infer_util import constrain_fn, find_valid_initial_params, init_to_uniform, potential_energy, transform_fn
from numpyro.util import cond, while_loop
AdaptWindow = namedtuple('AdaptWindow', ['start', 'end'])
AdaptState = namedtuple('AdaptState', ['step_size', 'inverse_mass_matrix', 'mass_matrix_sqrt',
'ss_state', 'mm_state', 'window_idx', 'rng'])
IntegratorState = namedtuple('IntegratorState', ['z', 'r', 'potential_energy', 'z_grad'])
TreeInfo = namedtuple('TreeInfo', ['z_left', 'r_left', 'z_left_grad',
'z_right', 'r_right', 'z_right_grad',
'z_proposal', 'z_proposal_pe', 'z_proposal_grad',
'depth', 'weight', 'r_sum', 'turning', 'diverging',
'sum_accept_probs', 'num_proposals'])
def dual_averaging(t0=10, kappa=0.75, gamma=0.05):
"""
Dual Averaging is a scheme to solve convex optimization problems. It
belongs to a class of subgradient methods which uses subgradients (which
lie in a dual space) to update states (in primal space) of a model. Under
some conditions, the averages of generated parameters during the scheme are
guaranteed to converge to an optimal value. However, a counter-intuitive
aspect of traditional subgradient methods is "new subgradients enter the
model with decreasing weights" (see reference [1]). Dual Averaging scheme
resolves that issue by updating parameters using weights equally for
subgradients, hence we have the name "dual averaging".
This class implements a dual averaging scheme which is adapted for Markov
chain Monte Carlo (MCMC) algorithms. To be more precise, we will replace
subgradients by some statistics calculated at the end of MCMC trajectories.
Following [2], we introduce some free parameters such as ``t0`` and
``kappa``, which is helpful and still guarantees the convergence of the
scheme.
**References:**
1. *Primal-dual subgradient methods for convex problems*,
Yurii Nesterov
2. *The No-U-turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo*,
Matthew D. Hoffman, Andrew Gelman
:param int t0: A free parameter introduced in reference [2] that stabilizes
the initial steps of the scheme. Defaults to 10.
:param float kappa: A free parameter introduced in reference [2] that
controls the weights of steps of the scheme. For a small ``kappa``, the
scheme will quickly forget states from early steps. This should be a
number in :math:`(0.5, 1]`. Defaults to 0.75.
:param float gamma: A free parameter introduced in reference [1] which
controls the speed of the convergence of the scheme. Defaults to 0.05.
:return: a (`init_fn`, `update_fn`) pair.
"""
def init_fn(prox_center=0.):
"""
:param float prox_center: A parameter introduced in reference [1] which
pulls the primal sequence towards it. Defaults to 0.
:return: initial state for the scheme.
"""
x_t = 0.
x_avg = 0. # average of primal sequence
g_avg = 0. # average of dual sequence
t = 0
return x_t, x_avg, g_avg, t, prox_center
def update_fn(g, state):
"""
:param float g: The current subgradient or statistics calculated during
an MCMC trajectory.
:param state: Current state of the scheme.
:return: new state for the scheme.
"""
x_t, x_avg, g_avg, t, prox_center = state
t = t + 1
# g_avg = (g_1 + ... + g_t) / t
g_avg = (1 - 1 / (t + t0)) * g_avg + g / (t + t0)
# According to formula (3.4) of [1], we have
# x_t = argmin{ g_avg . x + loc_t . |x - x0|^2 },
# hence x_t = x0 - g_avg / (2 * loc_t),
# where loc_t := beta_t / t, beta_t := (gamma/2) * sqrt(t).
x_t = prox_center - (t ** 0.5) / gamma * g_avg
# weight for the new x_t
weight_t = t ** (-kappa)
x_avg = (1 - weight_t) * x_avg + weight_t * x_t
return x_t, x_avg, g_avg, t, prox_center
return init_fn, update_fn
def welford_covariance(diagonal=True):
"""
Implements Welford's online method for estimating (co)variance. Useful for
adapting diagonal and dense mass structures for HMC. It is required that
each sample is a 1-dimensional array.
**References:**
1. *The Art of Computer Programming*,
Donald E. Knuth
:param bool diagonal: If True, we estimate the variance of samples.
Otherwise, we estimate the covariance of the samples. Defaults to True.
:return: a (`init_fn`, `update_fn`, `final_fn`) triple.
"""
def init_fn(size):
"""
:param int size: size of each sample.
:return: initial state for the scheme.
"""
mean = np.zeros(size)
if diagonal:
m2 = np.zeros(size)
else:
m2 = np.zeros((size, size))
n = 0
return mean, m2, n
def update_fn(sample, state):
"""
:param sample: A new sample.
:param state: Current state of the scheme.
:return: new state for the scheme.
"""
mean, m2, n = state
n = n + 1
delta_pre = sample - mean
mean = mean + delta_pre / n
delta_post = sample - mean
if diagonal:
m2 = m2 + delta_pre * delta_post
else:
m2 = m2 + np.outer(delta_post, delta_pre)
return mean, m2, n
def final_fn(state, regularize=False):
"""
:param state: Current state of the scheme.
:param bool regularize: Whether to adjust diagonal for numerical stability.
:return: a pair of estimated covariance and the square root of precision.
"""
mean, m2, n = state
# XXX it is not necessary to check for the case n=1
cov = m2 / (n - 1)
if regularize:
# Regularization from Stan
scaled_cov = (n / (n + 5)) * cov
shrinkage = 1e-3 * (5 / (n + 5))
if diagonal:
cov = scaled_cov + shrinkage
else:
cov = scaled_cov + shrinkage * np.identity(mean.shape[0])
if np.ndim(cov) == 2:
cov_inv_sqrt = cholesky_inverse(cov)
else:
cov_inv_sqrt = np.sqrt(np.reciprocal(cov))
return cov, cov_inv_sqrt
return init_fn, update_fn, final_fn
def velocity_verlet(potential_fn, kinetic_fn):
r"""
Second order symplectic integrator that uses the velocity verlet algorithm
for position `z` and momentum `r`.
:param potential_fn: Python callable that computes the potential energy
given input parameters. The input parameters to `potential_fn` can be
any python collection type.
:param kinetic_fn: Python callable that returns the kinetic energy given
inverse mass matrix and momentum.
:return: a pair of (`init_fn`, `update_fn`).
"""
def init_fn(z, r):
"""
:param z: Position of the particle.
:param r: Momentum of the particle.
:return: initial state for the integrator.
"""
potential_energy, z_grad = value_and_grad(potential_fn)(z)
return IntegratorState(z, r, potential_energy, z_grad)
def update_fn(step_size, inverse_mass_matrix, state):
"""
:param float step_size: Size of a single step.
:param inverse_mass_matrix: Inverse of mass matrix, which is used to
calculate kinetic energy.
:param state: Current state of the integrator.
:return: new state for the integrator.
"""
z, r, _, z_grad = state
r = tree_multimap(lambda r, z_grad: r - 0.5 * step_size * z_grad, r, z_grad) # r(n+1/2)
r_grad = grad(kinetic_fn, argnums=1)(inverse_mass_matrix, r)
z = tree_multimap(lambda z, r_grad: z + step_size * r_grad, z, r_grad) # z(n+1)
potential_energy, z_grad = value_and_grad(potential_fn)(z)
r = tree_multimap(lambda r, z_grad: r - 0.5 * step_size * z_grad, r, z_grad) # r(n+1)
return IntegratorState(z, r, potential_energy, z_grad)
return init_fn, update_fn
def find_reasonable_step_size(potential_fn, kinetic_fn, momentum_generator, inverse_mass_matrix,
position, rng, init_step_size):
"""
Finds a reasonable step size by tuning `init_step_size`. This function is used
to avoid working with a too large or too small step size in HMC.
**References:**
1. *The No-U-Turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo*,
Matthew D. Hoffman, Andrew Gelman
:param potential_fn: A callable to compute potential energy.
:param kinetic_fn: A callable to compute kinetic energy.
:param momentum_generator: A generator to get a random momentum variable.
:param inverse_mass_matrix: Inverse of mass matrix.
:param position: Current position of the particle.
:param jax.random.PRNGKey rng: Random key to be used as the source of randomness.
:param float init_step_size: Initial step size to be tuned.
:return: a reasonable value for step size.
:rtype: float
"""
# We are going to find a step_size which make accept_prob (Metropolis correction)
# near the target_accept_prob. If accept_prob:=exp(-delta_energy) is small,
# then we have to decrease step_size; otherwise, increase step_size.
target_accept_prob = np.log(0.8)
_, vv_update = velocity_verlet(potential_fn, kinetic_fn)
z = position
potential_energy, z_grad = value_and_grad(potential_fn)(z)
tiny = np.finfo(get_dtype(init_step_size)).tiny
def _body_fn(state):
step_size, _, direction, rng = state
rng, rng_momentum = random.split(rng)
# scale step_size: increase 2x or decrease 2x depends on direction;
# direction=1 means keep increasing step_size, otherwise decreasing step_size.
# Note that the direction is -1 if delta_energy is `NaN`, which may be the
# case for a diverging trajectory (e.g. in the case of evaluating log prob
# of a value simulated using a large step size for a constrained sample site).
step_size = (2.0 ** direction) * step_size
r = momentum_generator(inverse_mass_matrix, rng_momentum)
_, r_new, potential_energy_new, _ = vv_update(step_size,
inverse_mass_matrix,
(z, r, potential_energy, z_grad))
energy_current = kinetic_fn(inverse_mass_matrix, r) + potential_energy
energy_new = kinetic_fn(inverse_mass_matrix, r_new) + potential_energy_new
delta_energy = energy_new - energy_current
direction_new = np.where(target_accept_prob < -delta_energy, 1, -1)
return step_size, direction, direction_new, rng
def _cond_fn(state):
step_size, last_direction, direction, _ = state
# condition to run only if step_size is not so small or we are not decreasing step_size
not_small_step_size_cond = (step_size > tiny) | (direction >= 0)
return not_small_step_size_cond & ((last_direction == 0) | (direction == last_direction))
step_size, _, _, _ = while_loop(_cond_fn, _body_fn, (init_step_size, 0, 0, rng))
return step_size
def build_adaptation_schedule(num_steps):
"""
Builds a window adaptation schedule to be used during warmup phase of HMC.
:param int num_steps: Number of warmup steps.
:return: a list of contiguous windows, each has attributes `start` and `end`,
where `start` is the starting index and `end` is the ending index of the window.
**References:**
1. *Stan Reference Manual version 2.18*,
Stan Development Team
"""
adaptation_schedule = []
# from Stan, for small num_steps
if num_steps < 20:
adaptation_schedule.append(AdaptWindow(0, num_steps - 1))
return adaptation_schedule
# We separate num_steps into windows:
# start_buffer + window 1 + window 2 + window 3 + ... + end_buffer
# where the length of each window will be doubled for the next window.
# We won't adapt mass matrix during start and end buffers; and mass
# matrix will be updated at the end of each window. This is helpful
# for dealing with the intense computation of sampling momentum from the
# inverse of mass matrix.
start_buffer_size = 75 # from Stan
end_buffer_size = 50 # from Stan
init_window_size = 25 # from Stan
if (start_buffer_size + end_buffer_size + init_window_size) > num_steps:
start_buffer_size = int(0.15 * num_steps)
end_buffer_size = int(0.1 * num_steps)
init_window_size = num_steps - start_buffer_size - end_buffer_size
adaptation_schedule.append(AdaptWindow(start=0, end=start_buffer_size - 1))
end_window_start = num_steps - end_buffer_size
next_window_size = init_window_size
next_window_start = start_buffer_size
while next_window_start < end_window_start:
cur_window_start, cur_window_size = next_window_start, next_window_size
# Ensure that slow adaptation windows are monotonically increasing
if 3 * cur_window_size <= end_window_start - cur_window_start:
next_window_size = 2 * cur_window_size
else:
cur_window_size = end_window_start - cur_window_start
next_window_start = cur_window_start + cur_window_size
adaptation_schedule.append(AdaptWindow(cur_window_start, next_window_start - 1))
adaptation_schedule.append(AdaptWindow(end_window_start, num_steps - 1))
return adaptation_schedule
def _identity_step_size(inverse_mass_matrix, z, rng, step_size):
return step_size
def warmup_adapter(num_adapt_steps, find_reasonable_step_size=_identity_step_size,
adapt_step_size=True, adapt_mass_matrix=True,
dense_mass=False, target_accept_prob=0.8):
"""
A scheme to adapt tunable parameters, namely step size and mass matrix, during
the warmup phase of HMC.
:param int num_adapt_steps: Number of warmup steps.
:param find_reasonable_step_size: A callable to find a reasonable step size
at the beginning of each adaptation window.
:param bool adapt_step_size: A flag to decide if we want to adapt step_size
during warm-up phase using Dual Averaging scheme (defaults to ``True``).
:param bool adapt_mass_matrix: A flag to decide if we want to adapt mass
matrix during warm-up phase using Welford scheme (defaults to ``True``).
:param bool dense_mass: A flag to decide if mass matrix is dense or
diagonal (defaults to ``False``).
:param float target_accept_prob: Target acceptance probability for step size
adaptation using Dual Averaging. Increasing this value will lead to a smaller
step size, hence the sampling will be slower but more robust. Default to 0.8.
:return: a pair of (`init_fn`, `update_fn`).
"""
ss_init, ss_update = dual_averaging()
mm_init, mm_update, mm_final = welford_covariance(diagonal=not dense_mass)
adaptation_schedule = np.array(build_adaptation_schedule(num_adapt_steps))
num_windows = len(adaptation_schedule)
def init_fn(z, rng, step_size=1.0, inverse_mass_matrix=None, mass_matrix_size=None):
"""
:param z: Initial position of the integrator.
:param jax.random.PRNGKey rng: Random key to be used as the source of randomness.
:param float step_size: Initial step size.
:param inverse_mass_matrix: Inverse of the initial mass matrix. If ``None``,
inverse of mass matrix will be an identity matrix with size is decided
by the argument `mass_matrix_size`.
:param int mass_matrix_size: Size of the mass matrix.
:return: initial state of the adapt scheme.
"""
rng, rng_ss = random.split(rng)
if inverse_mass_matrix is None:
assert mass_matrix_size is not None
if dense_mass:
inverse_mass_matrix = np.identity(mass_matrix_size)
else:
inverse_mass_matrix = np.ones(mass_matrix_size)
mass_matrix_sqrt = inverse_mass_matrix
else:
if dense_mass:
mass_matrix_sqrt = cholesky_inverse(inverse_mass_matrix)
else:
mass_matrix_sqrt = np.sqrt(np.reciprocal(inverse_mass_matrix))
if adapt_step_size:
step_size = find_reasonable_step_size(inverse_mass_matrix, z, rng_ss, step_size)
ss_state = ss_init(np.log(10 * step_size))
mm_state = mm_init(inverse_mass_matrix.shape[-1])
window_idx = 0
return AdaptState(step_size, inverse_mass_matrix, mass_matrix_sqrt,
ss_state, mm_state, window_idx, rng)
def _update_at_window_end(z, rng_ss, state):
step_size, inverse_mass_matrix, mass_matrix_sqrt, ss_state, mm_state, window_idx, rng = state
if adapt_mass_matrix:
inverse_mass_matrix, mass_matrix_sqrt = mm_final(mm_state, regularize=True)
mm_state = mm_init(inverse_mass_matrix.shape[-1])
if adapt_step_size:
step_size = find_reasonable_step_size(inverse_mass_matrix, z, rng_ss, step_size)
ss_state = ss_init(np.log(10 * step_size))
return AdaptState(step_size, inverse_mass_matrix, mass_matrix_sqrt,
ss_state, mm_state, window_idx, rng)
def update_fn(t, accept_prob, z, state):
"""
:param int t: The current time step.
:param float accept_prob: Acceptance probability of the current trajectory.
:param z: New position drawn at the end of the current trajectory.
:param state: Current state of the adapt scheme.
:return: new state of the adapt scheme.
"""
step_size, inverse_mass_matrix, mass_matrix_sqrt, ss_state, mm_state, window_idx, rng = state
rng, rng_ss = random.split(rng)
# update step size state
if adapt_step_size:
ss_state = ss_update(target_accept_prob - accept_prob, ss_state)
# note: at the end of warmup phase, use average of log step_size
log_step_size, log_step_size_avg, *_ = ss_state
step_size = np.where(t == (num_adapt_steps - 1),
np.exp(log_step_size_avg),
np.exp(log_step_size))
# account the the case log_step_size is a so small negative number
step_size = np.clip(step_size, a_min=np.finfo(get_dtype(step_size)).tiny)
# update mass matrix state
is_middle_window = (0 < window_idx) & (window_idx < (num_windows - 1))
if adapt_mass_matrix:
z_flat, _ = ravel_pytree(z)
mm_state = cond(is_middle_window,
(z_flat, mm_state), lambda args: mm_update(*args),
mm_state, lambda x: x)
t_at_window_end = t == adaptation_schedule[window_idx, 1]
window_idx = np.where(t_at_window_end, window_idx + 1, window_idx)
state = AdaptState(step_size, inverse_mass_matrix, mass_matrix_sqrt,
ss_state, mm_state, window_idx, rng)
state = cond(t_at_window_end & is_middle_window,
(z, rng_ss, state), lambda args: _update_at_window_end(*args),
state, lambda x: x)
return state
return init_fn, update_fn
def _is_turning(inverse_mass_matrix, r_left, r_right, r_sum):
r_left, _ = ravel_pytree(r_left)
r_right, _ = ravel_pytree(r_right)
r_sum, _ = ravel_pytree(r_sum)
if inverse_mass_matrix.ndim == 2:
v_left = np.matmul(inverse_mass_matrix, r_left)
v_right = np.matmul(inverse_mass_matrix, r_right)
elif inverse_mass_matrix.ndim == 1:
v_left = np.multiply(inverse_mass_matrix, r_left)
v_right = np.multiply(inverse_mass_matrix, r_right)
# This implements dynamic termination criterion (ref [2], section A.4.2).
turning_at_left = np.dot(v_left, r_sum - r_left) <= 0
turning_at_right = np.dot(v_right, r_sum - r_right) <= 0
return turning_at_left | turning_at_right
def _uniform_transition_kernel(current_tree, new_tree):
# This function computes transition prob for subtrees (ref [2], section A.3.1).
# e^new_weight / (e^new_weight + e^current_weight)
transition_prob = expit(new_tree.weight - current_tree.weight)
return transition_prob
def _biased_transition_kernel(current_tree, new_tree):
# This function computes transition prob for main trees (ref [2], section A.3.2).
transition_prob = np.exp(new_tree.weight - current_tree.weight)
# If new tree is turning or diverging, we won't move the proposal
# to the new tree.
transition_prob = np.where(new_tree.turning | new_tree.diverging,
0.0, np.clip(transition_prob, a_max=1.0))
return transition_prob
def _combine_tree(current_tree, new_tree, inverse_mass_matrix, going_right, rng, biased_transition):
# Now we combine the current tree and the new tree. Note that outside
# leaves of the combined tree are determined by the direction.
z_left, r_left, z_left_grad, z_right, r_right, r_right_grad = cond(
going_right,
(current_tree, new_tree),
lambda trees: (trees[0].z_left, trees[0].r_left,
trees[0].z_left_grad, trees[1].z_right,
trees[1].r_right, trees[1].z_right_grad),
(new_tree, current_tree),
lambda trees: (trees[0].z_left, trees[0].r_left,
trees[0].z_left_grad, trees[1].z_right,
trees[1].r_right, trees[1].z_right_grad)
)
r_sum = tree_multimap(np.add, current_tree.r_sum, new_tree.r_sum)
if biased_transition:
transition_prob = _biased_transition_kernel(current_tree, new_tree)
turning = new_tree.turning | _is_turning(inverse_mass_matrix, r_left, r_right, r_sum)
else:
transition_prob = _uniform_transition_kernel(current_tree, new_tree)
turning = current_tree.turning
transition = random.bernoulli(rng, transition_prob)
z_proposal, z_proposal_pe, z_proposal_grad = cond(
transition,
new_tree, lambda tree: (tree.z_proposal, tree.z_proposal_pe, tree.z_proposal_grad),
current_tree, lambda tree: (tree.z_proposal, tree.z_proposal_pe, tree.z_proposal_grad)
)
tree_depth = current_tree.depth + 1
tree_weight = np.logaddexp(current_tree.weight, new_tree.weight)
diverging = new_tree.diverging
sum_accept_probs = current_tree.sum_accept_probs + new_tree.sum_accept_probs
num_proposals = current_tree.num_proposals + new_tree.num_proposals
return TreeInfo(z_left, r_left, z_left_grad, z_right, r_right, r_right_grad,
z_proposal, z_proposal_pe, z_proposal_grad,
tree_depth, tree_weight, r_sum, turning, diverging,
sum_accept_probs, num_proposals)
def _build_basetree(vv_update, kinetic_fn, z, r, z_grad, inverse_mass_matrix, step_size, going_right,
energy_current, max_delta_energy):
step_size = np.where(going_right, step_size, -step_size)
z_new, r_new, potential_energy_new, z_new_grad = vv_update(
step_size,
inverse_mass_matrix,
(z, r, energy_current, z_grad),
)
energy_new = potential_energy_new + kinetic_fn(inverse_mass_matrix, r_new)
delta_energy = energy_new - energy_current
# Handles the NaN case.
delta_energy = np.where(np.isnan(delta_energy), np.inf, delta_energy)
tree_weight = -delta_energy
diverging = delta_energy > max_delta_energy
accept_prob = np.clip(np.exp(-delta_energy), a_max=1.0)
return TreeInfo(z_new, r_new, z_new_grad, z_new, r_new, z_new_grad,
z_new, potential_energy_new, z_new_grad,
depth=0, weight=tree_weight, r_sum=r_new, turning=False,
diverging=diverging, sum_accept_probs=accept_prob, num_proposals=1)
def _get_leaf(tree, going_right):
return cond(going_right,
tree,
lambda tree: (tree.z_right, tree.r_right, tree.z_right_grad),
tree,
lambda tree: (tree.z_left, tree.r_left, tree.z_left_grad))
def _double_tree(current_tree, vv_update, kinetic_fn, inverse_mass_matrix, step_size,
going_right, rng, energy_current, max_delta_energy, r_ckpts, r_sum_ckpts):
key, transition_key = random.split(rng)
# If we are going to the right, start from the right leaf of the current tree.
z, r, z_grad = _get_leaf(current_tree, going_right)
new_tree = _iterative_build_subtree(current_tree.depth, vv_update, kinetic_fn,
z, r, z_grad, inverse_mass_matrix, step_size,
going_right, key, energy_current, max_delta_energy,
r_ckpts, r_sum_ckpts)
return _combine_tree(current_tree, new_tree, inverse_mass_matrix, going_right, transition_key,
True)
def _leaf_idx_to_ckpt_idxs(n):
# computes the number of non-zero bits except the last bit
# e.g. 6 -> 2, 7 -> 2, 13 -> 2
_, idx_max = while_loop(lambda nc: nc[0] > 0,
lambda nc: (nc[0] >> 1, nc[1] + (nc[0] & 1)),
(n >> 1, 0))
# computes the number of contiguous last non-zero bits
# e.g. 6 -> 0, 7 -> 3, 13 -> 1
_, num_subtrees = while_loop(lambda nc: (nc[0] & 1) != 0,
lambda nc: (nc[0] >> 1, nc[1] + 1),
(n, 0))
# TODO: explore the potential of setting idx_min=0 to allow more turning checks
# It will be useful in case: e.g. assume a tree 0 -> 7 is a circle,
# subtrees 0 -> 3, 4 -> 7 are half-circles, which two leaves might not
# satisfy turning condition;
# the full tree 0 -> 7 is a circle, which two leaves might also not satisfy
# turning condition;
# however, we can check the turning condition of the subtree 0 -> 5, which
# likely satisfies turning condition because its trajectory 3/4 of a circle.
idx_min = idx_max - num_subtrees + 1
return idx_min, idx_max
def _is_iterative_turning(inverse_mass_matrix, r, r_sum, r_ckpts, r_sum_ckpts, idx_min, idx_max):
def _body_fn(state):
i, _ = state
subtree_r_sum = r_sum - r_sum_ckpts[i] + r_ckpts[i]
return i - 1, _is_turning(inverse_mass_matrix, r_ckpts[i], r, subtree_r_sum)
_, turning = while_loop(lambda it: (it[0] >= idx_min) & ~it[1],
_body_fn,
(idx_max, False))
return turning
def _iterative_build_subtree(depth, vv_update, kinetic_fn, z, r, z_grad,
inverse_mass_matrix, step_size, going_right, rng,
energy_current, max_delta_energy, r_ckpts, r_sum_ckpts):
max_num_proposals = 2 ** depth
def _cond_fn(state):
tree, turning, _, _, _ = state
return (tree.num_proposals < max_num_proposals) & ~turning & ~tree.diverging
def _body_fn(state):
current_tree, _, r_ckpts, r_sum_ckpts, rng = state
rng, transition_rng = random.split(rng)
z, r, z_grad = _get_leaf(current_tree, going_right)
new_leaf = _build_basetree(vv_update, kinetic_fn, z, r, z_grad, inverse_mass_matrix, step_size,
going_right, energy_current, max_delta_energy)
new_tree = _combine_tree(current_tree, new_leaf, inverse_mass_matrix, going_right,
transition_rng, False)
leaf_idx = current_tree.num_proposals
ckpt_idx_min, ckpt_idx_max = _leaf_idx_to_ckpt_idxs(leaf_idx)
r, _ = ravel_pytree(new_leaf.r_right)
r_sum, _ = ravel_pytree(new_tree.r_sum)
# we update checkpoints when leaf_idx is even
r_ckpts, r_sum_ckpts = cond(leaf_idx % 2 == 0,
(r_ckpts, r_sum_ckpts),
lambda x: (index_update(x[0], ckpt_idx_max, r),
index_update(x[1], ckpt_idx_max, r_sum)),
(r_ckpts, r_sum_ckpts),
lambda x: x)
turning = _is_iterative_turning(inverse_mass_matrix, r, r_sum, r_ckpts, r_sum_ckpts,
ckpt_idx_min, ckpt_idx_max)
return new_tree, turning, r_ckpts, r_sum_ckpts, rng
basetree = _build_basetree(vv_update, kinetic_fn, z, r, z_grad, inverse_mass_matrix, step_size,
going_right, energy_current, max_delta_energy)
r_init, _ = ravel_pytree(basetree.r_left)
r_ckpts = index_update(r_ckpts, 0, r_init)
r_sum_ckpts = index_update(r_sum_ckpts, 0, r_init)
tree, turning, _, _, _ = while_loop(
_cond_fn,
_body_fn,
(basetree, False, r_ckpts, r_sum_ckpts, rng)
)
# update depth and turning condition
return TreeInfo(tree.z_left, tree.r_left, tree.z_left_grad,
tree.z_right, tree.r_right, tree.z_right_grad,
tree.z_proposal, tree.z_proposal_pe, tree.z_proposal_grad,
depth, tree.weight, tree.r_sum, turning, tree.diverging,
tree.sum_accept_probs, tree.num_proposals)
def build_tree(verlet_update, kinetic_fn, verlet_state, inverse_mass_matrix, step_size, rng,
max_delta_energy=1000., max_tree_depth=10):
"""
Builds a binary tree from the `verlet_state`. This is used in NUTS sampler.
**References:**
1. *The No-U-Turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo*,
Matthew D. Hoffman, Andrew Gelman
2. *A Conceptual Introduction to Hamiltonian Monte Carlo*,
Michael Betancourt
:param verlet_update: A callable to get a new integrator state given a current
integrator state.
:param kinetic_fn: A callable to compute kinetic energy.
:param verlet_state: Initial integrator state.
:param inverse_mass_matrix: Inverse of the mass matrix.
:param float step_size: Step size for the current trajectory.
:param jax.random.PRNGKey rng: random key to be used as the source of
randomness.
:param float max_delta_energy: A threshold to decide if the new state diverges
(based on the energy difference) too much from the initial integrator state.
:return: information of the tree.
:rtype: :data:`TreeInfo`
"""
z, r, potential_energy, z_grad = verlet_state
energy_current = potential_energy + kinetic_fn(inverse_mass_matrix, r)
r_ckpts = np.zeros((max_tree_depth, inverse_mass_matrix.shape[-1]))
r_sum_ckpts = np.zeros((max_tree_depth, inverse_mass_matrix.shape[-1]))
tree = TreeInfo(z, r, z_grad, z, r, z_grad, z, potential_energy, z_grad,
depth=0, weight=0., r_sum=r, turning=False, diverging=False,
sum_accept_probs=0., num_proposals=0)
def _cond_fn(state):
tree, _ = state
return (tree.depth < max_tree_depth) & ~tree.turning & ~tree.diverging
def _body_fn(state):
tree, key = state
key, direction_key, doubling_key = random.split(key, 3)
going_right = random.bernoulli(direction_key)
tree = _double_tree(tree, verlet_update, kinetic_fn, inverse_mass_matrix, step_size,
going_right, doubling_key, energy_current, max_delta_energy,
r_ckpts, r_sum_ckpts)
return tree, key
state = (tree, rng)
tree, _ = while_loop(_cond_fn, _body_fn, state)
return tree
def euclidean_kinetic_energy(inverse_mass_matrix, r):
r, _ = ravel_pytree(r)
if inverse_mass_matrix.ndim == 2:
v = np.matmul(inverse_mass_matrix, r)
elif inverse_mass_matrix.ndim == 1:
v = np.multiply(inverse_mass_matrix, r)
return 0.5 * np.dot(v, r)
[docs]def initialize_model(rng, model, *model_args, init_strategy=init_to_uniform, **model_kwargs):
"""
Given a model with Pyro primitives, returns a function which, given
unconstrained parameters, evaluates the potential energy (negative
joint density). In addition, this also returns initial parameters
sampled from the prior to initiate MCMC sampling and functions to
transform unconstrained values at sample sites to constrained values
within their respective support.
:param jax.random.PRNGKey rng: random number generator seed to
sample from the prior. The returned `init_params` will have the
batch shape ``rng.shape[:-1]``.
:param model: Python callable containing Pyro primitives.
:param `*model_args`: args provided to the model.
:param callable init_strategy: a per-site initialization function.
:param `**model_kwargs`: kwargs provided to the model.
:return: tuple of (`init_params`, `potential_fn`, `constrain_fn`),
`init_params` are values from the prior used to initiate MCMC,
`constrain_fn` is a callable that uses inverse transforms
to convert unconstrained HMC samples to constrained values that
lie within the site's support.
"""
seeded_model = seed(model, rng if rng.ndim == 1 else rng[0])
model_trace = trace(seeded_model).get_trace(*model_args, **model_kwargs)
constrained_values, inv_transforms = {}, {}
has_transformed_dist = False
for k, v in model_trace.items():
if v['type'] == 'sample' and not v['is_observed']:
if v['intermediates']:
constrained_values[k] = v['intermediates'][0][0]
inv_transforms[k] = biject_to(v['fn'].base_dist.support)
has_transformed_dist = True
else:
constrained_values[k] = v['value']
inv_transforms[k] = biject_to(v['fn'].support)
elif v['type'] == 'param':
constraint = v['kwargs'].pop('constraint', real)
transform = biject_to(constraint)
if isinstance(transform, ComposeTransform):
base_transform = transform.parts[0]
constrained_values[k] = base_transform(transform.inv(v['value']))
inv_transforms[k] = base_transform
has_transformed_dist = True
else:
inv_transforms[k] = transform
constrained_values[k] = v['value']
prototype_params = transform_fn(inv_transforms,
{k: v for k, v in constrained_values.items()},
invert=True)
# NB: we use model instead of seeded_model to prevent unexpected behaviours (if any)
potential_fn = jax.partial(potential_energy, model, model_args, model_kwargs, inv_transforms)
if has_transformed_dist:
# FIXME: why using seeded_model here triggers an error for funnel reparam example
# if we use MCMC class (mcmc function works fine)
constrain_fun = jax.partial(constrain_fn, model, model_args, model_kwargs, inv_transforms)
else:
constrain_fun = jax.partial(transform_fn, inv_transforms)
def single_chain_init(key):
return find_valid_initial_params(key, model, *model_args, init_strategy=init_strategy,
param_as_improper=True, prototype_params=prototype_params,
**model_kwargs)
if rng.ndim == 1:
init_params, is_valid = single_chain_init(rng)
else:
init_params, is_valid = lax.map(single_chain_init, rng)
if isinstance(is_valid, jax.interpreters.xla.DeviceArray):
if device_get(~np.all(is_valid)):
raise RuntimeError("Cannot find valid initial parameters. Please check your model again.")
return init_params, potential_fn, constrain_fun
[docs]def consensus(subposteriors, num_draws=None, diagonal=False, rng=None):
"""
Merges subposteriors following consensus Monte Carlo algorithm.
**References:**
1. *Bayes and big data: The consensus Monte Carlo algorithm*,
Steven L. Scott, Alexander W. Blocker, Fernando V. Bonassi, Hugh A. Chipman,
Edward I. George, Robert E. McCulloch
:param list subposteriors: a list in which each element is a collection of samples.
:param int num_draws: number of draws from the merged posterior.
:param bool diagonal: whether to compute weights using variance or covariance, defaults to
`False` (using covariance).
:param jax.random.PRNGKey rng: source of the randomness, defaults to `jax.random.PRNGKey(0)`.
:return: if `num_draws` is None, merges subposteriors without resampling; otherwise, returns
a collection of `num_draws` samples with the same data structure as each subposterior.
"""
# stack subposteriors
joined_subposteriors = tree_multimap(lambda *args: np.stack(args), *subposteriors)
# shape of joined_subposteriors: n_subs x n_samples x sample_shape
joined_subposteriors = vmap(vmap(lambda sample: ravel_pytree(sample)[0]))(joined_subposteriors)
if num_draws is not None:
rng = random.PRNGKey(0) if rng is None else rng
# randomly gets num_draws from subposteriors
n_subs = len(subposteriors)
n_samples = tree_flatten(subposteriors[0])[0][0].shape[0]
# shape of draw_idxs: n_subs x num_draws x sample_shape
draw_idxs = random.randint(rng, shape=(n_subs, num_draws), minval=0, maxval=n_samples)
joined_subposteriors = vmap(lambda x, idx: x[idx])(joined_subposteriors, draw_idxs)
if diagonal:
# compute weights for each subposterior (ref: Section 3.1 of [1])
weights = vmap(lambda x: 1 / np.var(x, ddof=1, axis=0))(joined_subposteriors)
normalized_weights = weights / np.sum(weights, axis=0)
# get weighted samples
samples_flat = np.einsum('ij,ikj->kj', normalized_weights, joined_subposteriors)
else:
weights = vmap(lambda x: np.linalg.inv(np.cov(x.T)))(joined_subposteriors)
normalized_weights = np.matmul(np.linalg.inv(np.sum(weights, axis=0)), weights)
samples_flat = np.einsum('ijk,ilk->lj', normalized_weights, joined_subposteriors)
# unravel_fn acts on 1 sample of a subposterior
_, unravel_fn = ravel_pytree(tree_map(lambda x: x[0], subposteriors[0]))
return vmap(lambda x: unravel_fn(x))(samples_flat)
[docs]def parametric(subposteriors, diagonal=False):
"""
Merges subposteriors following (embarrassingly parallel) parametric Monte Carlo algorithm.
**References:**
1. *Asymptotically Exact, Embarrassingly Parallel MCMC*,
Willie Neiswanger, Chong Wang, Eric Xing
:param list subposteriors: a list in which each element is a collection of samples.
:param bool diagonal: whether to compute weights using variance or covariance, defaults to
`False` (using covariance).
:return: the estimated mean and variance/covariance parameters of the joined posterior
"""
joined_subposteriors = tree_multimap(lambda *args: np.stack(args), *subposteriors)
joined_subposteriors = vmap(vmap(lambda sample: ravel_pytree(sample)[0]))(joined_subposteriors)
submeans = np.mean(joined_subposteriors, axis=1)
if diagonal:
# NB: jax.numpy.var does not support ddof=1, so we do it manually
weights = vmap(lambda x: 1 / np.var(x, ddof=1, axis=0))(joined_subposteriors)
var = 1 / np.sum(weights, axis=0)
normalized_weights = var * weights
# comparing to consensus implementation, we compute weighted mean here
mean = np.einsum('ij,ij->j', normalized_weights, submeans)
return mean, var
else:
weights = vmap(lambda x: np.linalg.inv(np.cov(x.T)))(joined_subposteriors)
cov = np.linalg.inv(np.sum(weights, axis=0))
normalized_weights = np.matmul(cov, weights)
# comparing to consensus implementation, we compute weighted mean here
mean = np.einsum('ijk,ik->j', normalized_weights, submeans)
return mean, cov
[docs]def parametric_draws(subposteriors, num_draws, diagonal=False, rng=None):
"""
Merges subposteriors following (embarrassingly parallel) parametric Monte Carlo algorithm.
**References:**
1. *Asymptotically Exact, Embarrassingly Parallel MCMC*,
Willie Neiswanger, Chong Wang, Eric Xing
:param list subposteriors: a list in which each element is a collection of samples.
:param int num_draws: number of draws from the merged posterior.
:param bool diagonal: whether to compute weights using variance or covariance, defaults to
`False` (using covariance).
:param jax.random.PRNGKey rng: source of the randomness, defaults to `jax.random.PRNGKey(0)`.
:return: a collection of `num_draws` samples with the same data structure as each subposterior.
"""
rng = random.PRNGKey(0) if rng is None else rng
if diagonal:
mean, var = parametric(subposteriors, diagonal=True)
samples_flat = dist.Normal(mean, np.sqrt(var)).sample(rng, (num_draws,))
else:
mean, cov = parametric(subposteriors, diagonal=False)
samples_flat = dist.MultivariateNormal(mean, cov).sample(rng, (num_draws,))
_, unravel_fn = ravel_pytree(tree_map(lambda x: x[0], subposteriors[0]))
return vmap(lambda x: unravel_fn(x))(samples_flat)