Merge branch 'standard-multivariate-normal' into 'master'

MultivariateGaussianDist(): use "standard" multivariate normal distribution internally

See merge request !1142

AUTHORS.md

@@ -24,6 +24,7 @@ Eric Thrane
 Ethan Payne
 Francisco Javier Hernandez
 Gregory Ashton
+Hank Hua
 Hector Estelles
 Ignacio Magaña Hernandez
 Isobel Marguarethe Romero-Shaw
@@ -34,6 +35,7 @@ Jeremy G Baier
 John Veitch
 Joshua Brandt
 Josh Willis
+Karl Wette
 Katerina Chatziioannou
 Kaylee de Soto
 Khun Sang Phukon

bilby/core/prior/joint.py

@@ -382,6 +382,7 @@ class MultivariateGaussianDist(BaseJointPriorDist):
         self.covs = []
         self.corrcoefs = []
         self.sigmas = []
+        self.logprodsigmas = []   # log of product of sigmas, needed for "standard" multivariate normal
         self.weights = []
         self.eigvalues = []
         self.eigvectors = []
@@ -528,6 +529,9 @@ class MultivariateGaussianDist(BaseJointPriorDist):
             self.covs.append(np.eye(self.num_vars))
             self.sigmas.append(np.ones(self.num_vars))
 
+        # compute log of product of sigmas, needed for "standard" multivariate normal
+        self.logprodsigmas.append(np.log(np.prod(self.sigmas[-1])))
+
         # get eigen values and vectors
         try:
             evals, evecs = np.linalg.eig(self.corrcoefs[-1])
@@ -557,9 +561,16 @@ class MultivariateGaussianDist(BaseJointPriorDist):
         # add the mode
         self.nmodes += 1
 
-        # add multivariate Gaussian
+        # add "standard" multivariate normal distribution
+        # - when the typical scales of the parameters are very different,
+        #   multivariate_normal() may complain that the covariance matrix is singular
+        # - instead pass zero means and correlation matrix instead of covariance matrix
+        #   to get the equivalent of a standard normal distribution in higher dimensions
+        # - this modifies the multivariate normal PDF as follows:
+        #     multivariate_normal(mean=mus, cov=cov).logpdf(x)
+        #     = multivariate_normal(mean=0, cov=corrcoefs).logpdf((x - mus)/sigmas) - logprodsigmas
         self.mvn.append(
-            scipy.stats.multivariate_normal(mean=self.mus[-1], cov=self.covs[-1])
+            scipy.stats.multivariate_normal(mean=np.zeros(self.num_vars), cov=self.corrcoefs[-1])
         )
 
     def _rescale(self, samp, **kwargs):
@@ -630,7 +641,9 @@ class MultivariateGaussianDist(BaseJointPriorDist):
         for j in range(samp.shape[0]):
             # loop over the modes and sum the probabilities
             for i in range(self.nmodes):
-                lnprob[j] = np.logaddexp(lnprob[j], self.mvn[i].logpdf(samp[j]))
+                # self.mvn[i] is a "standard" multivariate normal distribution; see add_mode()
+                z = (samp[j] - self.mus[i]) / self.sigmas[i]
+                lnprob[j] = np.logaddexp(lnprob[j], self.mvn[i].logpdf(z) - self.logprodsigmas[i])
 
         # set out-of-bounds values to -inf
         lnprob[outbounds] = -np.inf

test/core/prior/joint_test.py

@@ -39,5 +39,61 @@ MultivariateGaussianDist(
                 )
 
 
+class TestMultivariateGaussianDistParameterScales(unittest.TestCase):
+    def _test_mvg_ln_prob_diff_expected(self, mvg, mus, sigmas, corrcoefs):
+        # the columns of the Cholesky decompsition give the directions along which
+        # the multivariate Gaussian PDF will decrease by exact differences per unit
+        # sigma; test that these are as expected
+        ln_prob_mus = mvg.ln_prob(mus)
+        d = np.linalg.cholesky(corrcoefs)
+        for i in np.ndindex(4, 4, 4):
+            ln_prob_mus_sigmas_d_i = mvg.ln_prob(mus + sigmas * (d @ i))
+            diff_ln_prob = ln_prob_mus - ln_prob_mus_sigmas_d_i
+            diff_ln_prob_expected = 0.5 * np.sum(np.array(i)**2)
+            self.assertTrue(
+                np.allclose(diff_ln_prob, diff_ln_prob_expected)
+            )
+
+    def test_mvg_unit_scales(self):
+        # test using order-unity standard deviations and correlations
+        sigmas = 0.3 * np.ones(3)
+        corrcoefs = np.identity(3)
+        mus = np.array([3, 1, 2])
+        mvg = bilby.core.prior.MultivariateGaussianDist(
+            names=['a', 'b', 'c'],
+            mus=mus,
+            sigmas=sigmas,
+            corrcoefs=corrcoefs,
+        )
+
+        self._test_mvg_ln_prob_diff_expected(mvg, mus, sigmas, corrcoefs)
+
+    def test_mvg_cw_scales(self):
+        # test using standard deviations and correlations from the
+        # inverse Fisher information matrix for the frequency/spindown
+        # parameters of a continuous wave signal
+        T = 365.25 * 86400
+        rho = 10
+        sigmas = np.array([
+            5 * np.sqrt(3) / (2 * np.pi * T * rho),
+            6 * np.sqrt(5) / (np.pi * T**2 * rho),
+            60 * np.sqrt(7) / (np.pi * T**3 * rho)
+        ])
+        corrcoefs = np.identity(3)
+        corrcoefs[0, 2] = corrcoefs[2, 0] = -np.sqrt(21) / 5
+
+        # test MultivariateGaussianDist() can handle parameters with very different scales:
+        # - f ~ 100, fd ~ 1/T, fdd ~ 1/T^2
+        mus = [123.4, -5.6e-8, 9e-18]
+        mvg = bilby.core.prior.MultivariateGaussianDist(
+            names=["f", "fd", "fdd"],
+            mus=mus,
+            sigmas=sigmas,
+            corrcoefs=corrcoefs,
+        )
+
+        self._test_mvg_ln_prob_diff_expected(mvg, mus, sigmas, corrcoefs)
+
+
 if __name__ == "__main__":
     unittest.main()