Add DiscreteMarkovChain distribution (#100)

Author: jessegrabowski

docs/api_reference.rst

@@ -30,6 +30,7 @@ Distributions
 
    GenExtreme
    histogram_utils.histogram_approximation
+   DiscreteMarkovChain
 
 
 Gaussian Processess

notebooks/discrete_markov_chain.ipynb

@@ -18,7 +18,6 @@
 """
 
 from pymc_experimental.distributions.continuous import GenExtreme
+from pymc_experimental.distributions.timeseries import DiscreteMarkovChain
 
-__all__ = [
-    "GenExtreme",
-]
+__all__ = ["GenExtreme", "DiscreteMarkovChain"]

pymc_experimental/distributions/__init__.py

@@ -0,0 +1,267 @@
+import warnings
+from typing import List, Union
+
+import numpy as np
+import pymc as pm
+import pytensor
+import pytensor.tensor as pt
+from pymc.distributions.dist_math import check_parameters
+from pymc.distributions.distribution import (
+    Distribution,
+    SymbolicRandomVariable,
+    _moment,
+    moment,
+)
+from pymc.distributions.shape_utils import (
+    _change_dist_size,
+    change_dist_size,
+    get_support_shape_1d,
+)
+from pymc.logprob.abstract import _logprob
+from pymc.logprob.basic import logp
+from pymc.logprob.utils import ignore_logprob
+from pymc.pytensorf import intX
+from pymc.util import check_dist_not_registered
+from pytensor.graph.basic import Node
+from pytensor.tensor import TensorVariable
+from pytensor.tensor.random.op import RandomVariable
+
+
+def _make_outputs_info(n_lags: int, init_dist: Distribution) -> List[Union[Distribution, dict]]:
+    """
+    Two cases are needed for outputs_info in the scans used by DiscreteMarkovRv. If n_lags = 1, we need to throw away
+    the first dimension of init_dist_ or else markov_chain will have shape (steps, 1, *batch_size) instead of
+    desired (steps, *batch_size)
+
+    Parameters
+    ----------
+    n_lags: int
+        Number of lags the Markov Chain considers when transitioning to the next state
+    init_dist: RandomVariable
+        Distribution over initial states
+
+    Returns
+    -------
+    taps: list
+        Lags to be fed into pytensor.scan when drawing a markov chain
+    """
+
+    if n_lags > 1:
+        return [{"initial": init_dist, "taps": list(range(-n_lags, 0))}]
+    else:
+        return [init_dist[0]]
+
+
+class DiscreteMarkovChainRV(SymbolicRandomVariable):
+    n_lags: int
+    default_output = 1
+    _print_name = ("DiscreteMC", "\\operatorname{DiscreteMC}")
+
+    def __init__(self, *args, n_lags, **kwargs):
+        self.n_lags = n_lags
+        super().__init__(*args, **kwargs)
+
+    def update(self, node: Node):
+        return {node.inputs[-1]: node.outputs[0]}
+
+
+class DiscreteMarkovChain(Distribution):
+    r"""
+    A Discrete Markov Chain is a sequence of random variables
+
+    .. math::
+
+        \{x_t\}_{t=0}^T
+
+    Where transition probability :math:`P(x_t | x_{t-1})` depends only on the state of the system at :math:`x_{t-1}`.
+
+    Parameters
+    ----------
+    P: tensor
+        Matrix of transition probabilities between states. Rows must sum to 1.
+        One of P or P_logits must be provided.
+    P_logit: tensor, optional
+        Matrix of transition logits. Converted to probabilities via Softmax activation.
+        One of P or P_logits must be provided.
+    steps: tensor, optional
+        Length of the markov chain. Only needed if state is not provided.
+    init_dist : unnamed distribution, optional
+        Vector distribution for initial values. Unnamed refers to distributions
+        created with the ``.dist()`` API. Distribution should have shape n_states.
+        If not, it will be automatically resized.
+
+        .. warning:: init_dist will be cloned, rendering it independent of the one passed as input.
+
+    Notes
+    -----
+    The initial distribution will be cloned, rendering it distinct from the one passed as
+    input.
+
+    Examples
+    --------
+     Create a Markov Chain of length 100 with 3 states. The number of states is given by the shape of P,
+     3 in this case.
+
+    >>> with pm.Model() as markov_chain:
+    >>>     P = pm.Dirichlet("P", a=[1, 1, 1], size=(3,))
+    >>>     init_dist = pm.Categorical.dist(p = np.full(3, 1 / 3))
+    >>>     markov_chain = pm.DiscreteMarkovChain("markov_chain", P=P, init_dist=init_dist, shape=(100,))
+
+    """
+
+    rv_type = DiscreteMarkovChainRV
+
+    def __new__(cls, *args, steps=None, n_lags=1, **kwargs):
+        steps = get_support_shape_1d(
+            support_shape=steps,
+            shape=None,
+            dims=kwargs.get("dims", None),
+            observed=kwargs.get("observed", None),
+            support_shape_offset=n_lags,
+        )
+
+        return super().__new__(cls, *args, steps=steps, n_lags=n_lags, **kwargs)
+
+    @classmethod
+    def dist(cls, P=None, logit_P=None, steps=None, init_dist=None, n_lags=1, **kwargs):
+        steps = get_support_shape_1d(
+            support_shape=steps, shape=kwargs.get("shape", None), support_shape_offset=n_lags
+        )
+
+        if steps is None:
+            raise ValueError("Must specify steps or shape parameter")
+        if P is None and logit_P is None:
+            raise ValueError("Must specify P or logit_P parameter")
+        if P is not None and logit_P is not None:
+            raise ValueError("Must specify only one of either P or logit_P parameter")
+
+        if logit_P is not None:
+            P = pm.math.softmax(logit_P, axis=-1)
+
+        P = pt.as_tensor_variable(P)
+        steps = pt.as_tensor_variable(intX(steps))
+
+        if init_dist is not None:
+            if not isinstance(init_dist, TensorVariable) or not isinstance(
+                init_dist.owner.op, (RandomVariable, SymbolicRandomVariable)
+            ):
+                raise ValueError(
+                    f"Init dist must be a distribution created via the `.dist()` API, "
+                    f"got {type(init_dist)}"
+                )
+
+            check_dist_not_registered(init_dist)
+            if init_dist.owner.op.ndim_supp > 1:
+                raise ValueError(
+                    "Init distribution must have a scalar or vector support dimension, ",
+                    f"got ndim_supp={init_dist.owner.op.ndim_supp}.",
+                )
+        else:
+            warnings.warn(
+                "Initial distribution not specified, defaulting to "
+                "`Categorical.dist(p=pt.full((k_states, ), 1/k_states), shape=...)`. You can specify an init_dist "
+                "manually to suppress this warning.",
+                UserWarning,
+            )
+            k = P.shape[-1]
+            init_dist = pm.Categorical.dist(p=pt.full((k,), 1 / k))
+
+        # We can ignore init_dist, as it will be accounted for in the logp term
+        init_dist = ignore_logprob(init_dist)
+
+        return super().dist([P, steps, init_dist], n_lags=n_lags, **kwargs)
+
+    @classmethod
+    def rv_op(cls, P, steps, init_dist, n_lags, size=None):
+        if size is not None:
+            batch_size = size
+        else:
+            batch_size = pt.broadcast_shape(
+                P[tuple([...] + [0] * (n_lags + 1))], pt.atleast_1d(init_dist)[..., 0]
+            )
+
+        init_dist = change_dist_size(init_dist, (n_lags, *batch_size))
+        init_dist_ = init_dist.type()
+        P_ = P.type()
+        steps_ = steps.type()
+
+        state_rng = pytensor.shared(np.random.default_rng())
+
+        def transition(*args):
+            *states, transition_probs, old_rng = args
+            p = transition_probs[tuple(states)]
+            next_rng, next_state = pm.Categorical.dist(p=p, rng=old_rng).owner.outputs
+            return next_state, {old_rng: next_rng}
+
+        markov_chain, state_updates = pytensor.scan(
+            transition,
+            non_sequences=[P_, state_rng],
+            outputs_info=_make_outputs_info(n_lags, init_dist_),
+            n_steps=steps_,
+            strict=True,
+        )
+
+        (state_next_rng,) = tuple(state_updates.values())
+
+        discrete_mc_ = pt.moveaxis(pt.concatenate([init_dist_, markov_chain], axis=0), 0, -1)
+
+        discrete_mc_op = DiscreteMarkovChainRV(
+            inputs=[P_, steps_, init_dist_],
+            outputs=[state_next_rng, discrete_mc_],
+            ndim_supp=1,
+            n_lags=n_lags,
+        )
+
+        discrete_mc = discrete_mc_op(P, steps, init_dist)
+        return discrete_mc
+
+
+@_change_dist_size.register(DiscreteMarkovChainRV)
+def change_mc_size(op, dist, new_size, expand=False):
+    if expand:
+        old_size = dist.shape[:-1]
+        new_size = tuple(new_size) + tuple(old_size)
+
+    return DiscreteMarkovChain.rv_op(*dist.owner.inputs[:-1], size=new_size, n_lags=op.n_lags)
+
+
+@_moment.register(DiscreteMarkovChainRV)
+def discrete_mc_moment(op, rv, P, steps, init_dist, state_rng):
+    init_dist_moment = moment(init_dist)
+    n_lags = op.n_lags
+
+    def greedy_transition(*args):
+        *states, transition_probs, old_rng = args
+        p = transition_probs[tuple(states)]
+        return pt.argmax(p)
+
+    chain_moment, moment_updates = pytensor.scan(
+        greedy_transition,
+        non_sequences=[P, state_rng],
+        outputs_info=_make_outputs_info(n_lags, init_dist),
+        n_steps=steps,
+        strict=True,
+    )
+    chain_moment = pt.concatenate([init_dist_moment, chain_moment])
+    return chain_moment
+
+
+@_logprob.register(DiscreteMarkovChainRV)
+def discrete_mc_logp(op, values, P, steps, init_dist, state_rng, **kwargs):
+    value = values[0]
+    n_lags = op.n_lags
+
+    indexes = [value[..., i : -(n_lags - i) if n_lags != i else None] for i in range(n_lags + 1)]
+
+    mc_logprob = logp(init_dist, value[..., :n_lags]).sum(axis=-1)
+    mc_logprob += pt.log(P[tuple(indexes)]).sum(axis=-1)
+
+    return check_parameters(
+        mc_logprob,
+        pt.all(pt.eq(P.shape[-(n_lags + 1) :], P.shape[-1])),
+        pt.all(pt.allclose(P.sum(axis=-1), 1.0)),
+        pt.eq(pt.atleast_1d(init_dist).shape[0], n_lags),
+        msg="Last (n_lags + 1) dimensions of P must be square, "
+        "P must sum to 1 along the last axis, "
+        "First dimension of init_dist must be n_lags",
+    )