def _build_basetree(self, z, r, z_grads, log_slice, direction,
energy_current):
step_size = self.step_size if direction == 1 else -self.step_size
z_new, r_new, z_grads, potential_energy = velocity_verlet(
z,
r,
self.potential_fn,
self.inverse_mass_matrix,
step_size,
z_grads=z_grads)
r_new_flat = torch.cat(
[r_new[site_name].reshape(-1) for site_name in sorted(r_new)])
energy_new = potential_energy + self._kinetic_energy(r_new)
# handle the NaN case
energy_new = scalar_like(
energy_new,
float("inf")) if torch_isnan(energy_new) else energy_new
sliced_energy = energy_new + log_slice
diverging = (sliced_energy > self._max_sliced_energy)
delta_energy = energy_new - energy_current
accept_prob = (-delta_energy).exp().clamp(max=1.0)
if self.use_multinomial_sampling:
tree_weight = -sliced_energy
else:
# As a part of the slice sampling process (see below), along the trajectory
# we eliminate states which p(z, r) < u, or dE > 0.
# Due to this elimination (and stop doubling conditions),
# the weight of binary tree might not equal to 2^tree_depth.
tree_weight = scalar_like(sliced_energy,
1. if sliced_energy <= 0 else 0.)
return _TreeInfo(z_new, r_new, z_grads, z_new, r_new, z_grads, z_new,
potential_energy, z_grads, r_new_flat, tree_weight,
False, diverging, accept_prob, 1)
def sample(self, params):
z, potential_energy, z_grads = self._fetch_from_cache()
# recompute PE when cache is cleared
if z is None:
z = params
z_grads, potential_energy = potential_grad(self.potential_fn, z)
self._cache(z, potential_energy, z_grads)
# return early if no sample sites
elif len(z) == 0:
self._t += 1
self._mean_accept_prob = 1.
if self._t > self._warmup_steps:
self._accept_cnt += 1
return params
r, r_unscaled = self._sample_r(name="r_t={}".format(self._t))
energy_current = self._kinetic_energy(r_unscaled) + potential_energy
# Temporarily disable distributions args checking as
# NaNs are expected during step size adaptation
with optional(pyro.validation_enabled(False), self._t < self._warmup_steps):
z_new, r_new, z_grads_new, potential_energy_new = velocity_verlet(
z, r, self.potential_fn, self.mass_matrix_adapter.kinetic_grad,
self.step_size, self.num_steps, z_grads=z_grads)
# apply Metropolis correction.
r_new_unscaled = self.mass_matrix_adapter.unscale(r_new)
energy_proposal = self._kinetic_energy(r_new_unscaled) + potential_energy_new
delta_energy = energy_proposal - energy_current
# handle the NaN case which may be the case for a diverging trajectory
# when using a large step size.
delta_energy = scalar_like(delta_energy, float("inf")) if torch_isnan(delta_energy) else delta_energy
if delta_energy > self._max_sliced_energy and self._t >= self._warmup_steps:
self._divergences.append(self._t - self._warmup_steps)
accept_prob = (-delta_energy).exp().clamp(max=1.)
rand = pyro.sample("rand_t={}".format(self._t), dist.Uniform(scalar_like(accept_prob, 0.),
scalar_like(accept_prob, 1.)))
accepted = False
if rand < accept_prob:
accepted = True
z = z_new
z_grads = z_grads_new
self._cache(z, potential_energy_new, z_grads)
self._t += 1
if self._t > self._warmup_steps:
n = self._t - self._warmup_steps
if accepted:
self._accept_cnt += 1
else:
n = self._t
self._adapter.step(self._t, z, accept_prob, z_grads)
self._mean_accept_prob += (accept_prob.item() - self._mean_accept_prob) / n
return z.copy()
def _build_basetree(self, z, r, z_grads, log_slice, direction,
energy_current):
step_size = self.step_size if direction == 1 else -self.step_size
z_new, r_new, z_grads, potential_energy = velocity_verlet(
z,
r,
self.potential_fn,
self.mass_matrix_adapter.kinetic_grad,
step_size,
z_grads=z_grads,
)
r_new_unscaled = self.mass_matrix_adapter.unscale(r_new)
energy_new = potential_energy + self._kinetic_energy(r_new_unscaled)
# handle the NaN case
energy_new = (scalar_like(energy_new, float("inf"))
if torch_isnan(energy_new) else energy_new)
sliced_energy = energy_new + log_slice
diverging = sliced_energy > self._max_sliced_energy
delta_energy = energy_new - energy_current
accept_prob = (-delta_energy).exp().clamp(max=1.0)
if self.use_multinomial_sampling:
tree_weight = -sliced_energy
else:
# As a part of the slice sampling process (see below), along the trajectory
# we eliminate states which p(z, r) < u, or dE > 0.
# Due to this elimination (and stop doubling conditions),
# the weight of binary tree might not equal to 2^tree_depth.
tree_weight = scalar_like(sliced_energy,
1.0 if sliced_energy <= 0 else 0.0)
r_sum = r_new_unscaled
return _TreeInfo(
z_new,
r_new,
r_new_unscaled,
z_grads,
z_new,
r_new,
r_new_unscaled,
z_grads,
z_new,
potential_energy,
z_grads,
r_sum,
tree_weight,
False,
diverging,
accept_prob,
1,
)
def not_pooled(at_bats, hits):
r"""
Number of hits in $K$ at bats for each player has a Binomial
distribution with independent probability of success, $\phi_i$.
:param (torch.Tensor) at_bats: Number of at bats for each player.
:param (torch.Tensor) hits: Number of hits for the given at bats.
:return: Number of hits predicted by the model.
"""
num_players = at_bats.shape[0]
with pyro.plate("num_players", num_players):
phi_prior = Uniform(scalar_like(at_bats, 0), scalar_like(at_bats, 1))
phi = pyro.sample("phi", phi_prior)
return pyro.sample("obs", Binomial(at_bats, phi), obs=hits)
def partially_pooled_with_logit(at_bats, hits):
r"""
Number of hits has a Binomial distribution with a logit link function.
The logits $\alpha$ for each player is normally distributed with the
mean and scale parameters sharing a common prior.
:param (torch.Tensor) at_bats: Number of at bats for each player.
:param (torch.Tensor) hits: Number of hits for the given at bats.
:return: Number of hits predicted by the model.
"""
num_players = at_bats.shape[0]
loc = pyro.sample("loc", Normal(scalar_like(at_bats, -1), scalar_like(at_bats, 1)))
scale = pyro.sample("scale", HalfCauchy(scale=scalar_like(at_bats, 1)))
with pyro.plate("num_players", num_players):
alpha = pyro.sample("alpha", Normal(loc, scale))
return pyro.sample("obs", Binomial(at_bats, logits=alpha), obs=hits)
def partially_pooled(at_bats, hits):
r"""
Number of hits has a Binomial distribution with independent
probability of success, $\phi_i$. Each $\phi_i$ follows a Beta
distribution with concentration parameters $c_1$ and $c_2$, where
$c_1 = m * kappa$, $c_2 = (1 - m) * kappa$, $m ~ Uniform(0, 1)$,
and $kappa ~ Pareto(1, 1.5)$.
:param (torch.Tensor) at_bats: Number of at bats for each player.
:param (torch.Tensor) hits: Number of hits for the given at bats.
:return: Number of hits predicted by the model.
"""
num_players = at_bats.shape[0]
m = pyro.sample("m", Uniform(scalar_like(at_bats, 0), scalar_like(at_bats, 1)))
kappa = pyro.sample("kappa", Pareto(scalar_like(at_bats, 1), scalar_like(at_bats, 1.5)))
with pyro.plate("num_players", num_players):
phi_prior = Beta(m * kappa, (1 - m) * kappa)
phi = pyro.sample("phi", phi_prior)
return pyro.sample("obs", Binomial(at_bats, phi), obs=hits)
def sample(self, params):
z, potential_energy, z_grads = self._fetch_from_cache()
# recompute PE when cache is cleared
if z is None:
z = params
potential_energy = self.potential_fn(z)
self._cache(z, potential_energy)
# return early if no sample sites
elif len(z) == 0:
self._t += 1
self._mean_accept_prob = 1.
if self._t > self._warmup_steps:
self._accept_cnt += 1
return z
r, r_flat = self._sample_r(name="r_t={}".format(self._t))
energy_current = self._kinetic_energy(r) + potential_energy
# Ideally, following a symplectic integrator trajectory, the energy is constant.
# In that case, we can sample the proposal uniformly, and there is no need to use "slice".
# However, it is not the case for real situation: there are errors during the computation.
# To deal with that problem, as in [1], we introduce an auxiliary "slice" variable (denoted
# by u).
# The sampling process goes as follows:
# first sampling u from initial state (z_0, r_0) according to
# u ~ Uniform(0, p(z_0, r_0)),
# then sampling state (z, r) from the integrator trajectory according to
# (z, r) ~ Uniform({(z', r') in trajectory | p(z', r') >= u}).
#
# For more information about slice sampling method, see [3].
# For another version of NUTS which uses multinomial sampling instead of slice sampling,
# see [2].
if self.use_multinomial_sampling:
log_slice = -energy_current
else:
# Rather than sampling the slice variable from `Uniform(0, exp(-energy))`, we can
# sample log_slice directly using `energy`, so as to avoid potential underflow or
# overflow issues ([2]).
slice_exp_term = pyro.sample("slicevar_exp_t={}".format(self._t),
dist.Exponential(scalar_like(energy_current, 1.)))
log_slice = -energy_current - slice_exp_term
z_left = z_right = z
r_left = r_right = r
z_left_grads = z_right_grads = z_grads
accepted = False
r_sum = r_flat
sum_accept_probs = 0.
num_proposals = 0
tree_weight = scalar_like(energy_current, 0. if self.use_multinomial_sampling else 1.)
# Temporarily disable distributions args checking as
# NaNs are expected during step size adaptation.
with optional(pyro.validation_enabled(False), self._t < self._warmup_steps):
# doubling process, stop when turning or diverging
tree_depth = 0
while tree_depth < self._max_tree_depth:
direction = pyro.sample("direction_t={}_treedepth={}".format(self._t, tree_depth),
dist.Bernoulli(probs=scalar_like(tree_weight, 0.5)))
direction = int(direction.item())
if direction == 1: # go to the right, start from the right leaf of current tree
new_tree = self._build_tree(z_right, r_right, z_right_grads, log_slice,
direction, tree_depth, energy_current)
# update leaf for the next doubling process
z_right = new_tree.z_right
r_right = new_tree.r_right
z_right_grads = new_tree.z_right_grads
else: # go the the left, start from the left leaf of current tree
new_tree = self._build_tree(z_left, r_left, z_left_grads, log_slice,
direction, tree_depth, energy_current)
z_left = new_tree.z_left
r_left = new_tree.r_left
z_left_grads = new_tree.z_left_grads
sum_accept_probs = sum_accept_probs + new_tree.sum_accept_probs
num_proposals = num_proposals + new_tree.num_proposals
# stop doubling
if new_tree.diverging:
if self._t >= self._warmup_steps:
self._divergences.append(self._t - self._warmup_steps)
break
if new_tree.turning:
break
tree_depth += 1
if self.use_multinomial_sampling:
new_tree_prob = (new_tree.weight - tree_weight).exp()
else:
new_tree_prob = new_tree.weight / tree_weight
rand = pyro.sample("rand_t={}_treedepth={}".format(self._t, tree_depth),
dist.Uniform(scalar_like(new_tree_prob, 0.),
scalar_like(new_tree_prob, 1.)))
if rand < new_tree_prob:
accepted = True
z = new_tree.z_proposal
self._cache(z, new_tree.z_proposal_pe, new_tree.z_proposal_grads)
r_sum = r_sum + new_tree.r_sum
if self._is_turning(r_left, r_right, r_sum): # stop doubling
break
else: # update tree_weight
if self.use_multinomial_sampling:
tree_weight = _logaddexp(tree_weight, new_tree.weight)
else:
tree_weight = tree_weight + new_tree.weight
accept_prob = sum_accept_probs / num_proposals
self._t += 1
if self._t > self._warmup_steps:
n = self._t - self._warmup_steps
if accepted:
self._accept_cnt += 1
else:
n = self._t
self._adapter.step(self._t, z, accept_prob)
self._mean_accept_prob += (accept_prob.item() - self._mean_accept_prob) / n
return z.copy()
def _build_tree(self, z, r, z_grads, log_slice, direction, tree_depth, energy_current):
if tree_depth == 0:
return self._build_basetree(z, r, z_grads, log_slice, direction, energy_current)
# build the first half of tree
half_tree = self._build_tree(z, r, z_grads, log_slice,
direction, tree_depth-1, energy_current)
z_proposal = half_tree.z_proposal
z_proposal_pe = half_tree.z_proposal_pe
z_proposal_grads = half_tree.z_proposal_grads
# Check conditions to stop doubling. If we meet that condition,
# there is no need to build the other tree.
if half_tree.turning or half_tree.diverging:
return half_tree
# Else, build remaining half of tree.
# If we are going to the right, start from the right leaf of the first half.
if direction == 1:
z = half_tree.z_right
r = half_tree.r_right
z_grads = half_tree.z_right_grads
else: # otherwise, start from the left leaf of the first half
z = half_tree.z_left
r = half_tree.r_left
z_grads = half_tree.z_left_grads
other_half_tree = self._build_tree(z, r, z_grads, log_slice,
direction, tree_depth-1, energy_current)
if self.use_multinomial_sampling:
tree_weight = _logaddexp(half_tree.weight, other_half_tree.weight)
else:
tree_weight = half_tree.weight + other_half_tree.weight
sum_accept_probs = half_tree.sum_accept_probs + other_half_tree.sum_accept_probs
num_proposals = half_tree.num_proposals + other_half_tree.num_proposals
r_sum = half_tree.r_sum + other_half_tree.r_sum
# The probability of that proposal belongs to which half of tree
# is computed based on the weights of each half.
if self.use_multinomial_sampling:
other_half_tree_prob = (other_half_tree.weight - tree_weight).exp()
else:
# For the special case that the weights of each half are both 0,
# we choose the proposal from the first half
# (any is fine, because the probability of picking it at the end is 0!).
other_half_tree_prob = (other_half_tree.weight / tree_weight if tree_weight > 0
else scalar_like(tree_weight, 0.))
is_other_half_tree = pyro.sample("is_other_half_tree",
dist.Bernoulli(probs=other_half_tree_prob))
if is_other_half_tree == 1:
z_proposal = other_half_tree.z_proposal
z_proposal_pe = other_half_tree.z_proposal_pe
z_proposal_grads = other_half_tree.z_proposal_grads
# leaves of the full tree are determined by the direction
if direction == 1:
z_left = half_tree.z_left
r_left = half_tree.r_left
z_left_grads = half_tree.z_left_grads
z_right = other_half_tree.z_right
r_right = other_half_tree.r_right
z_right_grads = other_half_tree.z_right_grads
else:
z_left = other_half_tree.z_left
r_left = other_half_tree.r_left
z_left_grads = other_half_tree.z_left_grads
z_right = half_tree.z_right
r_right = half_tree.r_right
z_right_grads = half_tree.z_right_grads
# We already check if first half tree is turning. Now, we check
# if the other half tree or full tree are turning.
turning = other_half_tree.turning or self._is_turning(r_left, r_right, r_sum)
# The divergence is checked by the second half tree (the first half is already checked).
diverging = other_half_tree.diverging
return _TreeInfo(z_left, r_left, z_left_grads, z_right, r_right, z_right_grads, z_proposal,
z_proposal_pe, z_proposal_grads, r_sum, tree_weight, turning, diverging,
sum_accept_probs, num_proposals)