Add ZeroSumNormal distribution (pyro-ppl#1751)

* added zerosumnormal and tests

* added edge case handling for support shape

* removed commented out functions

* added zerosumnormal to docs

* fixed zerosumnormal support shape default

* Added v1 of docstrings for zerosumnormal

* updated zsn docstring

* improved init shape handling for zerosumnormal

* improved docstrings

* added ZeroSumTransform

* made n_zerosum_axes an attribute for the zerosumtransform

* removed commented out lines

* added zerosumtransform class

* switched zsn from ParameterFreeTransform to Transform

* changed ZeroSumNormal to transformed distibutrion

* changed input to tuple for _transform_to_zero_sum

* added forward and inverse shape to transform, fixed zero_sum constraint handling

* fixed failing zsn tests

* added docstring, removed whitespace, fixed missing import

* fixed allclose to be assert allclose

* linted and formatted

* added sample code to docstring for zsn

* updated docstring

* removed list from ZeroSum constraint call

* removed unneeded iteration, updated docstring

* updated constraint code

* added ZeroSumTransform to docs

* fixed transform shapes

* added doctest example for zsn

* added constraint test

* added zero_sum constraint to docs

* added type hinting to transforms file

* fixed docs formatting

* moved skip zsn from test_gof earlier

* reversed zerosumtransform

* broadcasted mean and var of zsn

* added stricter zero_sum constraint tol, improved mean and var functions

* fixed _transform_to_zero_sum

* removed shape promote from zsn, changed broadcast to zeros_like

* chose better zsn test cases

* Update zero_sum constraint feasible_like

Co-authored-by: Till Hoffmann <[email protected]>

* fixed docstring for doctests

---------

Co-authored-by: Till Hoffmann <[email protected]>

docs/source/distributions.rst

@@ -380,6 +380,13 @@ Weibull
     :show-inheritance:
     :member-order: bysource
 
+ZeroSumNormal
+^^^^^^^^^^^^^
+.. autoclass:: numpyro.distributions.continuous.ZeroSumNormal
+    :members:
+    :undoc-members:
+    :show-inheritance:
+    :member-order: bysource
 
 Discrete Distributions
 ----------------------
@@ -820,6 +827,9 @@ unit_interval
 ^^^^^^^^^^^^^
 .. autodata:: numpyro.distributions.constraints.unit_interval
 
+zero_sum
+^^^^^^^^
+.. autodata:: numpyro.distributions.constraints.zero_sum
 
 Transforms
 ----------
@@ -1014,6 +1024,15 @@ StickBreakingTransform
     :show-inheritance:
     :member-order: bysource
 
+ZeroSumTransform
+^^^^^^^^^^^^^^^^
+
+.. autoclass:: numpyro.distributions.transforms.ZeroSumTransform
+    :members:
+    :undoc-members:
+    :show-inheritance:
+    :member-order: bysource
+
 
 Flows
 -----

numpyro/distributions/__init__.py

@@ -47,6 +47,7 @@
     StudentT,
     Uniform,
     Weibull,
+    ZeroSumNormal,
 )
 from numpyro.distributions.copula import GaussianCopula, GaussianCopulaBeta
 from numpyro.distributions.directional import (
@@ -196,4 +197,5 @@
     "ZeroInflatedDistribution",
     "ZeroInflatedPoisson",
     "ZeroInflatedNegativeBinomial2",
+    "ZeroSumNormal",
 ]

numpyro/distributions/constraints.py

@@ -55,6 +55,7 @@
     "softplus_lower_cholesky",
     "softplus_positive",
     "unit_interval",
+    "zero_sum",
     "Constraint",
 ]
 
@@ -697,6 +698,29 @@ def feasible_like(self, prototype):
         return jax.numpy.full_like(prototype, prototype.shape[-1] ** (-0.5))
 
 
+class _ZeroSum(Constraint):
+    def __init__(self, event_dim=1):
+        self.event_dim = event_dim
+        super().__init__()
+
+    def __call__(self, x):
+        jnp = np if isinstance(x, (np.ndarray, np.generic)) else jax.numpy
+        tol = jnp.finfo(x.dtype).eps * x.shape[-1] * 10
+        zerosum_true = True
+        for dim in range(-self.event_dim, 0):
+            zerosum_true = zerosum_true & jnp.allclose(x.sum(dim), 0, atol=tol)
+        return zerosum_true
+
+    def __eq__(self, other):
+        return type(self) is type(other) and self.event_dim == other.event_dim
+
+    def feasible_like(self, prototype):
+        return jax.numpy.zeros_like(prototype)
+
+    def tree_flatten(self):
+        return (self.event_dim,), (("event_dim",), dict())
+
+
 # TODO: Make types consistent
 # See https://github.com/pytorch/pytorch/issues/50616
 
@@ -731,3 +755,4 @@ def feasible_like(self, prototype):
 sphere = _Sphere()
 unit_interval = _UnitInterval()
 open_interval = _OpenInterval
+zero_sum = _ZeroSum

numpyro/distributions/continuous.py

@@ -58,6 +58,7 @@
     ExpTransform,
     PowerTransform,
     SigmoidTransform,
+    ZeroSumTransform,
 )
 from numpyro.distributions.util import (
     add_diag,
@@ -2438,3 +2439,97 @@ def cdf(self, value):
 
     def icdf(self, value):
         return self._ald.icdf(value)
+
+
+class ZeroSumNormal(TransformedDistribution):
+    r"""
+    Zero Sum Normal distribution adapted from PyMC [1] as described in [2,3]. This is a Normal distribution where one or
+    more axes are constrained to sum to zero (the last axis by default).
+
+    .. math::
+        \begin{align*}
+        ZSN(\sigma) = N(0, \sigma^2 (I - \tfrac{1}{n}J)) \\
+        \text{where} \ ~ J_{ij} = 1 \ ~ \text{and} \\
+        n = \text{number of zero-sum axes}
+        \end{align*}
+
+    :param array_like scale: Standard deviation of the underlying normal distribution before the zerosum constraint is
+        enforced.
+    :param tuple event_shape: The event shape of the distribution, the axes of which get constrained to sum to zero.
+
+    **Example:**
+
+    .. doctest::
+
+        >>> from numpy.testing import assert_allclose
+        >>> from jax import random
+        >>> import jax.numpy as jnp
+        >>> import numpyro
+        >>> import numpyro.distributions as dist
+        >>> from numpyro.infer import MCMC, NUTS
+
+        >>> N = 1000
+        >>> n_categories = 20
+        >>> rng_key = random.PRNGKey(0)
+        >>> key1, key2, key3 = random.split(rng_key, 3)
+        >>> category_ind = random.choice(key1, jnp.arange(n_categories), shape=(N,))
+        >>> beta = random.normal(key2, shape=(n_categories,))
+        >>> beta -= beta.mean(-1)
+        >>> y = 5 + beta[category_ind] + random.normal(key3, shape=(N,))
+
+        >>> def model(category_ind, y): # category_ind is an indexed categorical variable with 20 categories
+        ...     N = len(category_ind)
+        ...     alpha = numpyro.sample("alpha", dist.Normal(0, 2.5))
+        ...     beta = numpyro.sample("beta", dist.ZeroSumNormal(1, event_shape=(n_categories,)))
+        ...     sigma =  numpyro.sample("sigma", dist.Exponential(1))
+        ...     with numpyro.plate("observations", N):
+        ...         mu = alpha + beta[category_ind]
+        ...         obs = numpyro.sample("obs", dist.Normal(mu, sigma), obs=y)
+        ...     return obs
+
+        >>> nuts_kernel = NUTS(model=model, target_accept_prob=0.9)
+        >>> mcmc = MCMC(
+        ...     sampler=nuts_kernel,
+        ...     num_samples=1_000, num_warmup=1_000, num_chains=4
+        ... )
+        >>> mcmc.run(random.PRNGKey(0), category_ind=category_ind, y=y)
+        >>> posterior_samples = mcmc.get_samples()
+        >>> # Confirm everything along last axis sums to zero
+        >>> assert_allclose(posterior_samples['beta'].sum(-1), 0, atol=1e-3)
+
+    **References**
+    [1] https://github.com/pymc-devs/pymc/blob/6252d2e58dc211c913ee2e652a4058d271d48bbd/pymc/distributions/multivariate.py#L2637
+    [2] https://www.pymc.io/projects/docs/en/stable/api/distributions/generated/pymc.ZeroSumNormal.html
+    [3] https://learnbayesstats.com/episode/74-optimizing-nuts-developing-zerosumnormal-distribution-adrian-seyboldt/
+    """
+
+    arg_constraints = {"scale": constraints.positive}
+    reparametrized_params = ["scale"]
+
+    def __init__(self, scale, event_shape, *, validate_args=None):
+        event_ndim = len(event_shape)
+        transformed_shape = tuple(size - 1 for size in event_shape)
+        self.scale = scale
+        super().__init__(
+            Normal(0, scale).expand(transformed_shape).to_event(event_ndim),
+            ZeroSumTransform(event_ndim),
+            validate_args=validate_args,
+        )
+
+    @constraints.dependent_property(is_discrete=False)
+    def support(self):
+        return constraints.zero_sum(len(self.event_shape))
+
+    @property
+    def mean(self):
+        return jnp.zeros(self.batch_shape + self.event_shape)
+
+    @property
+    def variance(self):
+        event_ndim = len(self.event_shape)
+        zero_sum_axes = tuple(range(-event_ndim, 0))
+        theoretical_var = jnp.square(self.scale)
+        for axis in zero_sum_axes:
+            theoretical_var *= 1 - 1 / self.event_shape[axis]
+
+        return jnp.broadcast_to(theoretical_var, self.batch_shape + self.event_shape)

numpyro/distributions/transforms.py

@@ -6,6 +6,7 @@
 import weakref
 
 import numpy as np
+from numpy.core.numeric import normalize_axis_tuple
 
 from jax import lax, vmap
 from jax.flatten_util import ravel_pytree
@@ -50,6 +51,7 @@
     "StickBreakingTransform",
     "Transform",
     "UnpackTransform",
+    "ZeroSumTransform",
 ]
 
 
@@ -1380,6 +1382,92 @@ def __eq__(self, other):
         return jnp.array_equal(self.transition_matrix, other.transition_matrix)
 
 
+class ZeroSumTransform(Transform):
+    """A transform that constrains an array to sum to zero, adapted from PyMC [1] as described in [2,3]
+
+    :param transform_ndims: Number of trailing dimensions to transform.
+
+    **References**
+    [1] https://github.com/pymc-devs/pymc/blob/244fb97b01ad0f3dadf5c3837b65839e2a59a0e8/pymc/distributions/transforms.py#L266
+    [2] https://www.pymc.io/projects/docs/en/stable/api/distributions/generated/pymc.ZeroSumNormal.html
+    [3] https://learnbayesstats.com/episode/74-optimizing-nuts-developing-zerosumnormal-distribution-adrian-seyboldt/
+    """
+
+    def __init__(self, transform_ndims: int = 1) -> None:
+        self.transform_ndims = transform_ndims
+
+    @property
+    def domain(self) -> constraints.Constraint:
+        return constraints.independent(constraints.real, self.transform_ndims)
+
+    @property
+    def codomain(self) -> constraints.Constraint:
+        return constraints.zero_sum(self.transform_ndims)
+
+    def __call__(self, x: jnp.ndarray) -> jnp.ndarray:
+        zero_sum_axes = tuple(range(-self.transform_ndims, 0))
+        for axis in zero_sum_axes:
+            x = self.extend_axis(x, axis=axis)
+        return x
+
+    def _inverse(self, y: jnp.ndarray) -> jnp.ndarray:
+        zero_sum_axes = tuple(range(-self.transform_ndims, 0))
+        for axis in zero_sum_axes:
+            y = self.extend_axis_rev(y, axis=axis)
+        return y
+
+    def extend_axis_rev(self, array: jnp.ndarray, axis: int) -> jnp.ndarray:
+        normalized_axis = normalize_axis_tuple(axis, array.ndim)[0]
+
+        n = array.shape[normalized_axis]
+        last = jnp.take(array, jnp.array([-1]), axis=normalized_axis)
+
+        sum_vals = -last * jnp.sqrt(n)
+        norm = sum_vals / (jnp.sqrt(n) + n)
+        slice_before = (slice(None, None),) * normalized_axis
+        return array[(*slice_before, slice(None, -1))] + norm
+
+    def extend_axis(self, array: jnp.ndarray, axis: int) -> jnp.ndarray:
+        n = array.shape[axis] + 1
+
+        sum_vals = array.sum(axis, keepdims=True)
+        norm = sum_vals / (jnp.sqrt(n) + n)
+        fill_val = norm - sum_vals / jnp.sqrt(n)
+
+        out = jnp.concatenate([array, fill_val], axis=axis)
+        return out - norm
+
+    def log_abs_det_jacobian(
+        self, x: jnp.ndarray, y: jnp.ndarray, intermediates: None = None
+    ) -> jnp.ndarray:
+        shape = jnp.broadcast_shapes(
+            x.shape[: -self.transform_ndims], y.shape[: -self.transform_ndims]
+        )
+        return jnp.zeros_like(x, shape=shape)
+
+    def forward_shape(self, shape: tuple) -> tuple:
+        return shape[: -self.transform_ndims] + tuple(
+            s + 1 for s in shape[-self.transform_ndims :]
+        )
+
+    def inverse_shape(self, shape: tuple) -> tuple:
+        return shape[: -self.transform_ndims] + tuple(
+            s - 1 for s in shape[-self.transform_ndims :]
+        )
+
+    def tree_flatten(self):
+        aux_data = {
+            "transform_ndims": self.transform_ndims,
+        }
+        return (), ((), aux_data)
+
+    def __eq__(self, other):
+        return (
+            isinstance(other, ZeroSumTransform)
+            and self.transform_ndims == other.transform_ndims
+        )
+
+
 ##########################################################
 # CONSTRAINT_REGISTRY
 ##########################################################
@@ -1530,3 +1618,8 @@ def _transform_to_softplus_lower_cholesky(constraint):
 @biject_to.register(constraints.simplex)
 def _transform_to_simplex(constraint):
     return StickBreakingTransform()
+
+
+@biject_to.register(constraints.zero_sum)
+def _transform_to_zero_sum(constraint):
+    return ZeroSumTransform(constraint.event_dim)

test/test_constraints.py

@@ -62,6 +62,7 @@ class T(namedtuple("TestCase", ["constraint_cls", "params", "kwargs"])):
         dict(),
     ),
     "open_interval": T(constraints.open_interval, (_a(-1.0), _a(1.0)), dict()),
+    "zero_sum": T(constraints.zero_sum, (), dict(event_dim=1)),
 }
 
 # TODO: BijectorConstraint