Conditional autoregression distribution (#4504)

* Add conditional autoregression distribution (CAR)

Author: ckrapu

RELEASE-NOTES.md

@@ -6,6 +6,7 @@
 
 ### New Features
 + `pm.math.cartesian` can now handle inputs that are themselves >1D (see [#4482](https://github.com/pymc-devs/pymc3/pull/4482)).
++ The `CAR` distribution has been added to allow for use of conditional autoregressions which often are used in spatial and network models.
 + ...
 
 ### Maintenance

docs/source/api/distributions/multivariate.rst

@@ -15,6 +15,7 @@ Multivariate
    Multinomial
    Dirichlet
    DirichletMultinomial
+   CAR
 
 .. automodule:: pymc3.distributions.multivariate
    :members:

pymc3/distributions/__init__.py

@@ -80,6 +80,7 @@
 )
 from pymc3.distributions.mixture import Mixture, MixtureSameFamily, NormalMixture
 from pymc3.distributions.multivariate import (
+    CAR,
     Dirichlet,
     DirichletMultinomial,
     KroneckerNormal,
@@ -184,4 +185,5 @@
     "Simulator",
     "fast_sample_posterior_predictive",
     "BART",
+    "CAR",
 ]

pymc3/distributions/multivariate.py

@@ -66,6 +66,7 @@
     "LKJCholeskyCov",
     "MatrixNormal",
     "KroneckerNormal",
+    "CAR",
 ]
 
 
@@ -2089,3 +2090,127 @@ def logp(self, value):
 
     def _distr_parameters_for_repr(self):
         return ["mu"]
+
+
+class CAR(Continuous):
+    R"""
+    Likelihood for a conditional autoregression. This is a special case of the
+    multivariate normal with an adjacency-structured covariance matrix.
+
+    .. math::
+
+       f(x \mid W, \alpha, \tau) =
+           \frac{|T|^{1/2}}{(2\pi)^{k/2}}
+           \exp\left\{ -\frac{1}{2} (x-\mu)^{\prime} T^{-1} (x-\mu) \right\}
+
+    where :math:`T = (\tau D(I-\alpha W))^{-1}` and :math:`D = diag(\sum_i W_{ij})`.
+
+    ========  ==========================
+    Support   :math:`x \in \mathbb{R}^k`
+    Mean      :math:`\mu \in \mathbb{R}^k`
+    Variance  :math:`(\tau D(I-\alpha W))^{-1}`
+    ========  ==========================
+
+    Parameters
+    ----------
+    mu: array
+        Real-valued mean vector
+    W: Numpy matrix
+        Symmetric adjacency matrix of 1s and 0s indicating
+        adjacency between elements.
+    alpha: float or array
+        Autoregression parameter taking values between -1 and 1. Values closer to 0 indicate weaker
+        correlation and values closer to 1 indicate higher autocorrelation. For most use cases, the
+        support of alpha should be restricted to (0, 1)
+    tau: float or array
+        Positive precision variable controlling the scale of the underlying normal variates.
+    sparse: bool, default=False
+        Determines whether or not sparse computations are used
+
+    References
+    ----------
+    ..  Jin, X., Carlin, B., Banerjee, S.
+        "Generalized Hierarchical Multivariate CAR Models for Areal Data"
+        Biometrics, Vol. 61, No. 4 (Dec., 2005), pp. 950-961
+    """
+
+    def __init__(self, mu, W, alpha, tau, sparse=False, *args, **kwargs):
+        super().__init__(*args, **kwargs)
+
+        D = W.sum(axis=0)
+        d, _ = W.shape
+
+        self.d = d
+        self.median = self.mode = self.mean = self.mu = aet.as_tensor_variable(mu)
+        self.sparse = sparse
+
+        if not W.ndim == 2 or not np.allclose(W, W.T):
+            raise ValueError("W must be a symmetric adjacency matrix.")
+
+        if sparse:
+            W_sparse = scipy.sparse.csr_matrix(W)
+            self.W = aesara.sparse.as_sparse_variable(W_sparse)
+        else:
+            self.W = aet.as_tensor_variable(W)
+
+        # eigenvalues of D^−1/2 * W * D^−1/2
+        Dinv_sqrt = np.diag(1 / np.sqrt(D))
+        DWD = np.matmul(np.matmul(Dinv_sqrt, W), Dinv_sqrt)
+        self.lam = scipy.linalg.eigvalsh(DWD)
+        self.D = aet.as_tensor_variable(D)
+
+        tau = aet.as_tensor_variable(tau)
+        if tau.ndim > 0:
+            self.tau = tau[:, None]
+        else:
+            self.tau = tau
+
+        alpha = aet.as_tensor_variable(alpha)
+        if alpha.ndim > 0:
+            self.alpha = alpha[:, None]
+        else:
+            self.alpha = alpha
+
+    def logp(self, value):
+        """
+        Calculate log-probability of a CAR-distributed vector
+        at specified value. This log probability function differs from
+        the true CAR log density (AKA a multivariate normal with CAR-structured
+        covariance matrix) by an additive constant.
+
+        Parameters
+        ----------
+        value: array
+            Value for which log-probability is calculated.
+
+        Returns
+        -------
+        TensorVariable
+        """
+
+        if value.ndim == 1:
+            value = value[None, :]
+
+        logtau = self.d * aet.log(self.tau).sum()
+        logdet = aet.log(1 - self.alpha.T * self.lam[:, None]).sum()
+        delta = value - self.mu
+
+        if self.sparse:
+            Wdelta = aesara.sparse.dot(delta, self.W)
+        else:
+            Wdelta = aet.dot(delta, self.W)
+
+        tau_dot_delta = self.D[None, :] * delta - self.alpha * Wdelta
+        logquad = (self.tau * delta * tau_dot_delta).sum(axis=-1)
+        return bound(
+            0.5 * (logtau + logdet - logquad),
+            aet.all(self.alpha <= 1),
+            aet.all(self.alpha >= -1),
+            self.tau > 0,
+        )
+
+    def random(self, point=None, size=None):
+        raise NotImplementedError("Sampling from a CAR distribution is not supported.")
+
+    def _distr_parameters_for_repr(self):
+        return ["mu", "W", "alpha", "tau"]

pymc3/tests/test_distributions.py

@@ -37,6 +37,7 @@
 from pymc3.blocking import DictToVarBijection
 from pymc3.distributions import (
     AR1,
+    CAR,
     AsymmetricLaplace,
     Bernoulli,
     Beta,
@@ -2614,6 +2615,42 @@ def test_orderedlogistic_dimensions(shape):
     assert np.allclose(ologp, expected)
 
 
+@pytest.mark.parametrize("shape", [(4,), (4, 1), (4, 4)], ids=str)
+def test_car_logp(shape):
+    """
+    Tests the log probability function for the CAR distribution by checking
+    against Scipy's multivariate normal logpdf, up to an additive constant.
+    The formula used by the CAR logp implementation omits several additive terms.
+    """
+    np.random.seed(1)
+
+    xs = np.random.randn(*shape)
+
+    # d x d adjacency matrix for a square (d=4) of rook-adjacent sites
+    W = np.array(
+        [[0.0, 1.0, 1.0, 0.0], [1.0, 0.0, 0.0, 1.0], [1.0, 0.0, 0.0, 1.0], [0.0, 1.0, 1.0, 0.0]]
+    )
+
+    tau = 2.0
+    alpha = 0.5
+    mu = np.zeros(4)
+
+    # Compute CAR covariance matrix and resulting MVN logp
+    D = W.sum(axis=0)
+    prec = tau * (np.diag(D) - alpha * W)
+    cov = np.linalg.inv(prec)
+    scipy_logp = scipy.stats.multivariate_normal.logpdf(xs, mu, cov)
+
+    car_logp = CAR.dist(mu, W, alpha, tau, shape=shape).logp(xs).eval()
+
+    # Check to make sure that the CAR and MVN log PDFs are equivalent
+    # up to an additive constant which is independent of the CAR parameters
+    delta_logp = scipy_logp - car_logp
+
+    # Check to make sure all the delta values are identical.
+    assert np.allclose(delta_logp - delta_logp[0], 0.0)
+
+
 class TestBugfixes:
     @pytest.mark.parametrize(
         "dist_cls,kwargs", [(MvNormal, dict(mu=0)), (MvStudentT, dict(mu=0, nu=2))]