# Source code for pymc.step_methods.hmc.nuts

```
# Copyright 2023 The PyMC Developers
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
from __future__ import annotations
from collections import namedtuple
import numpy as np
from pymc.math import logbern
from pymc.pytensorf import floatX
from pymc.stats.convergence import SamplerWarning
from pymc.step_methods.compound import Competence
from pymc.step_methods.hmc import integration
from pymc.step_methods.hmc.base_hmc import BaseHMC, DivergenceInfo, HMCStepData
from pymc.step_methods.hmc.integration import IntegrationError, State
from pymc.vartypes import continuous_types
__all__ = ["NUTS"]
[docs]class NUTS(BaseHMC):
r"""A sampler for continuous variables based on Hamiltonian mechanics.
NUTS automatically tunes the step size and the number of steps per
sample. A detailed description can be found at [1], "Algorithm 6:
Efficient No-U-Turn Sampler with Dual Averaging".
NUTS provides a number of statistics that can be accessed with
`trace.get_sampler_stats`:
- `mean_tree_accept`: The mean acceptance probability for the tree
that generated this sample. The mean of these values across all
samples but the burn-in should be approximately `target_accept`
(the default for this is 0.8).
- `diverging`: Whether the trajectory for this sample diverged. If
there are any divergences after burnin, this indicates that
the results might not be reliable. Reparametrization can
often help, but you can also try to increase `target_accept` to
something like 0.9 or 0.95.
- `energy`: The energy at the point in phase-space where the sample
was accepted. This can be used to identify posteriors with
problematically long tails. See below for an example.
- `energy_change`: The difference in energy between the start and
the end of the trajectory. For a perfect integrator this would
always be zero.
- `max_energy_change`: The maximum difference in energy along the
whole trajectory.
- `depth`: The depth of the tree that was used to generate this sample
- `tree_size`: The number of leafs of the sampling tree, when the
sample was accepted. This is usually a bit less than
`2 ** depth`. If the tree size is large, the sampler is
using a lot of leapfrog steps to find the next sample. This can for
example happen if there are strong correlations in the posterior,
if the posterior has long tails, if there are regions of high
curvature ("funnels"), or if the variance estimates in the mass
matrix are inaccurate. Reparametrisation of the model or estimating
the posterior variances from past samples might help.
- `tune`: This is `True`, if step size adaptation was turned on when
this sample was generated.
- `step_size`: The step size used for this sample.
- `step_size_bar`: The current best known step-size. After the tuning
samples, the step size is set to this value. This should converge
during tuning.
- `model_logp`: The model log-likelihood for this sample.
- `process_time_diff`: The time it took to draw the sample, as defined
by the python standard library `time.process_time`. This counts all
the CPU time, including worker processes in BLAS and OpenMP.
- `perf_counter_diff`: The time it took to draw the sample, as defined
by the python standard library `time.perf_counter` (wall time).
- `perf_counter_start`: The value of `time.perf_counter` at the beginning
of the computation of the draw.
- `index_in_trajectory`: This is usually only interesting for debugging
purposes. This indicates the position of the posterior draw in the
trajectory. Eg a -4 would indicate that the draw was the result of the
fourth leapfrog step in negative direction.
- `largest_eigval` and `smallest_eigval`: Experimental statistics for
some mass matrix adaptation algorithms. This is nan if it is not used.
References
----------
.. [1] Hoffman, Matthew D., & Gelman, Andrew. (2011). The No-U-Turn
Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo.
"""
name = "nuts"
default_blocked = True
stats_dtypes_shapes = {
"depth": (np.int64, []),
"step_size": (np.float64, []),
"tune": (bool, []),
"mean_tree_accept": (np.float64, []),
"step_size_bar": (np.float64, []),
"tree_size": (np.float64, []),
"diverging": (bool, []),
"energy_error": (np.float64, []),
"energy": (np.float64, []),
"max_energy_error": (np.float64, []),
"model_logp": (np.float64, []),
"process_time_diff": (np.float64, []),
"perf_counter_diff": (np.float64, []),
"perf_counter_start": (np.float64, []),
"largest_eigval": (np.float64, []),
"smallest_eigval": (np.float64, []),
"index_in_trajectory": (np.int64, []),
"reached_max_treedepth": (bool, []),
"warning": (SamplerWarning, None),
}
[docs] def __init__(self, vars=None, max_treedepth=10, early_max_treedepth=8, **kwargs):
r"""Set up the No-U-Turn sampler.
Parameters
----------
vars: list, default=None
List of value variables. If None, all continuous RVs from the
model are included.
Emax: float, default 1000
Maximum energy change allowed during leapfrog steps. Larger
deviations will abort the integration.
target_accept: float, default .8
Adapt the step size such that the average acceptance
probability across the trajectories are close to target_accept.
Higher values for target_accept lead to smaller step sizes.
Setting this to higher values like 0.9 or 0.99 can help
with sampling from difficult posteriors. Valid values are
between 0 and 1 (exclusive).
step_scale: float, default 0.25
Size of steps to take, automatically scaled down by `1/n**(1/4)`.
If step size adaptation is switched off, the resulting step size
is used. If adaptation is enabled, it is used as initial guess.
gamma: float, default .05
k: float, default .75
Parameter for dual averaging for step size adaptation. Values
between 0.5 and 1 (exclusive) are admissible. Higher values
correspond to slower adaptation.
t0: int, default 10
Parameter for dual averaging. Higher values slow initial
adaptation.
adapt_step_size: bool, default=True
Whether step size adaptation should be enabled. If this is
disabled, `k`, `t0`, `gamma` and `target_accept` are ignored.
max_treedepth: int, default=10
The maximum tree depth. Trajectories are stopped when this
depth is reached.
early_max_treedepth: int, default=8
The maximum tree depth during the first 200 tuning samples.
scaling: array_like, ndim = {1,2}
The inverse mass, or precision matrix. One dimensional arrays are
interpreted as diagonal matrices. If `is_cov` is set to True,
this will be interpreted as the mass or covariance matrix.
is_cov: bool, default=False
Treat the scaling as mass or covariance matrix.
potential: Potential, optional
An object that represents the Hamiltonian with methods `velocity`,
`energy`, and `random` methods. It can be specified instead
of the scaling matrix.
model: pymc.Model
The model
kwargs: passed to BaseHMC
Notes
-----
The step size adaptation stops when `self.tune` is set to False.
This is usually achieved by setting the `tune` parameter if
`pm.sample` to the desired number of tuning steps.
"""
super().__init__(vars, **kwargs)
self.max_treedepth = max_treedepth
self.early_max_treedepth = early_max_treedepth
self._reached_max_treedepth = 0
def _hamiltonian_step(self, start, p0, step_size):
if self.tune and self.iter_count < 200:
max_treedepth = self.early_max_treedepth
else:
max_treedepth = self.max_treedepth
tree = _Tree(len(p0), self.integrator, start, step_size, self.Emax)
reached_max_treedepth = False
for _ in range(max_treedepth):
direction = logbern(np.log(0.5)) * 2 - 1
divergence_info, turning = tree.extend(direction)
if divergence_info or turning:
break
else:
reached_max_treedepth = not self.tune
stats = tree.stats()
accept_stat = stats["mean_tree_accept"]
stats["reached_max_treedepth"] = reached_max_treedepth
return HMCStepData(tree.proposal, accept_stat, divergence_info, stats)
[docs] @staticmethod
def competence(var, has_grad):
"""Check how appropriate this class is for sampling a random variable."""
if var.dtype in continuous_types and has_grad:
return Competence.PREFERRED
return Competence.INCOMPATIBLE
# A proposal for the next position
Proposal = namedtuple("Proposal", "q, q_grad, energy, logp, index_in_trajectory")
# A subtree of the binary tree built by nuts.
Subtree = namedtuple(
"Subtree",
"left, right, p_sum, proposal, log_size",
)
class _Tree:
def __init__(
self,
ndim: int,
integrator: integration.CpuLeapfrogIntegrator,
start: State,
step_size: float,
Emax: float,
):
"""Binary tree from the NUTS algorithm.
Parameters
----------
leapfrog: function
A function that performs a single leapfrog step.
start: integration.State
The starting point of the trajectory.
step_size: float
The step size to use in this tree
Emax: float
The maximum energy change to accept before aborting the
transition as diverging.
"""
self.ndim = ndim
self.integrator = integrator
self.start = start
self.step_size = step_size
self.Emax = Emax
self.start_energy = start.energy
self.left = self.right = start
self.proposal = Proposal(start.q.data, start.q_grad, start.energy, start.model_logp, 0)
self.depth = 0
self.log_size = 0.0
self.log_accept_sum = -np.inf
self.mean_tree_accept = 0.0
self.n_proposals = 0
self.p_sum = start.p.data.copy()
self.max_energy_change = 0.0
def extend(self, direction):
"""Double the treesize by extending the tree in the given direction.
If direction is larger than 0, extend it to the right, otherwise
extend it to the left.
Return a tuple `(diverging, turning)` of type (DivergenceInfo, bool).
`diverging` indicates, that the tree extension was aborted because
the energy change exceeded `self.Emax`. `turning` indicates that
the tree extension was stopped because the termination criterior
was reached (the trajectory is turning back).
"""
if direction > 0:
tree, diverging, turning = self._build_subtree(
self.right, self.depth, floatX(np.asarray(self.step_size))
)
leftmost_begin, leftmost_end = self.left, self.right
rightmost_begin, rightmost_end = tree.left, tree.right
leftmost_p_sum = self.p_sum.copy()
rightmost_p_sum = tree.p_sum
self.right = tree.right
else:
tree, diverging, turning = self._build_subtree(
self.left, self.depth, floatX(np.asarray(-self.step_size))
)
leftmost_begin, leftmost_end = tree.right, tree.left
rightmost_begin, rightmost_end = self.left, self.right
leftmost_p_sum = tree.p_sum
rightmost_p_sum = self.p_sum.copy()
self.left = tree.right
self.depth += 1
if diverging or turning:
return diverging, turning
size1, size2 = self.log_size, tree.log_size
if logbern(size2 - size1):
self.proposal = tree.proposal
self.log_size = np.logaddexp(self.log_size, tree.log_size)
self.p_sum[:] += tree.p_sum
# Additional turning check only when tree depth > 0 to avoid redundant work
if self.depth > 0:
left, right = self.left, self.right
p_sum = self.p_sum
turning = (p_sum.dot(left.v) <= 0) or (p_sum.dot(right.v) <= 0)
p_sum1 = leftmost_p_sum + rightmost_begin.p.data
turning1 = (p_sum1.dot(leftmost_begin.v) <= 0) or (p_sum1.dot(rightmost_begin.v) <= 0)
p_sum2 = leftmost_end.p.data + rightmost_p_sum
turning2 = (p_sum2.dot(leftmost_end.v) <= 0) or (p_sum2.dot(rightmost_end.v) <= 0)
turning = turning | turning1 | turning2
return diverging, turning
def _single_step(self, left: State, epsilon: float):
"""Perform a leapfrog step and handle error cases."""
right: State | None
error: IntegrationError | None
error_msg: str | None
try:
right = self.integrator.step(epsilon, left)
except IntegrationError as err:
error_msg = str(err)
error = err
right = None
else:
assert right is not None # since there was no IntegrationError
# h - H0
energy_change = right.energy - self.start_energy
if np.isnan(energy_change):
energy_change = np.inf
self.log_accept_sum = np.logaddexp(self.log_accept_sum, min(0, -energy_change))
if np.abs(energy_change) > np.abs(self.max_energy_change):
self.max_energy_change = energy_change
if energy_change < self.Emax:
# Acceptance statistic
# e^{H(q_0, p_0) - H(q_n, p_n)} max(1, e^{H(q_0, p_0) - H(q_n, p_n)})
# Saturated Metropolis accept probability with Boltzmann weight
log_size = -energy_change
proposal = Proposal(
right.q.data,
right.q_grad,
right.energy,
right.model_logp,
right.index_in_trajectory,
)
tree = Subtree(right, right, right.p.data, proposal, log_size)
return tree, None, False
else:
error_msg = f"Energy change in leapfrog step is too large: {energy_change}."
error = None
finally:
self.n_proposals += 1
tree = Subtree(None, None, None, None, -np.inf)
divergence_info = DivergenceInfo(error_msg, error, left, right)
return tree, divergence_info, False
def _build_subtree(self, left, depth, epsilon):
if depth == 0:
return self._single_step(left, epsilon)
tree1, diverging, turning = self._build_subtree(left, depth - 1, epsilon)
if diverging or turning:
return tree1, diverging, turning
tree2, diverging, turning = self._build_subtree(tree1.right, depth - 1, epsilon)
left, right = tree1.left, tree2.right
if not (diverging or turning):
p_sum = tree1.p_sum + tree2.p_sum
turning = (p_sum.dot(left.v) <= 0) or (p_sum.dot(right.v) <= 0)
# Additional U turn check only when depth > 1 to avoid redundant work.
if depth - 1 > 0:
p_sum1 = tree1.p_sum + tree2.left.p.data
turning1 = (p_sum1.dot(tree1.left.v) <= 0) or (p_sum1.dot(tree2.left.v) <= 0)
p_sum2 = tree1.right.p.data + tree2.p_sum
turning2 = (p_sum2.dot(tree1.right.v) <= 0) or (p_sum2.dot(tree2.right.v) <= 0)
turning = turning | turning1 | turning2
log_size = np.logaddexp(tree1.log_size, tree2.log_size)
if logbern(tree2.log_size - log_size):
proposal = tree2.proposal
else:
proposal = tree1.proposal
else:
p_sum = tree1.p_sum
log_size = tree1.log_size
proposal = tree1.proposal
tree = Subtree(left, right, p_sum, proposal, log_size)
return tree, diverging, turning
def stats(self):
self.mean_tree_accept = np.exp(self.log_accept_sum) / self.n_proposals
return {
"depth": self.depth,
"mean_tree_accept": self.mean_tree_accept,
"energy_error": self.proposal.energy - self.start.energy,
"energy": self.proposal.energy,
"tree_size": self.n_proposals,
"max_energy_error": self.max_energy_change,
"model_logp": self.proposal.logp,
"index_in_trajectory": self.proposal.index_in_trajectory,
}
```