Add LogExpM1 transformation (#2601)

* Add softplus transformation

Softplus transformation (from non-negative to reals) might be more numerically stable (see Fig. 9 in Kucukelbir et al. 2017).

* add test

* Change name and implement a more numerically stable logexpm1

See https://github.com/tensorflow/tensorflow/blob/0b0d3c12ace80381f4a44365d30275a9a262609b/tensorflow/python/ops/distributions/util.py#L1009 for the derivation

* change default transformation for PositiveContinuous

* Revert "change default transformation for PositiveContinuous"

This reverts commit 8bc036c.

* name change

pymc3/distributions/transforms.py

@@ -8,7 +8,7 @@
 from .distribution import draw_values
 import numpy as np
 
-__all__ = ['transform', 'stick_breaking', 'logodds', 'interval',
+__all__ = ['transform', 'stick_breaking', 'logodds', 'interval', 'log_exp_m1',
            'lowerbound', 'upperbound', 'log', 'sum_to_1', 't_stick_breaking']
 
 
@@ -105,12 +105,31 @@ def jacobian_det(self, x):
 log = Log()
 
 
+class LogExpM1(ElemwiseTransform):
+    name = "log_exp_m1"
+
+    def backward(self, x):
+        return tt.nnet.softplus(x)
+
+    def forward(self, x):
+        """Inverse operation of softplus
+        y = Log(Exp(x) - 1) 
+          = Log(1 - Exp(-x)) + x
+        """
+        return tt.log(1.-tt.exp(-x)) + x
+
+    def forward_val(self, x, point=None):
+        return self.forward(x)
+
+    def jacobian_det(self, x):
+        return -tt.nnet.softplus(-x)
+
+log_exp_m1 = LogExpM1()
+
+
 class LogOdds(ElemwiseTransform):
     name = "logodds"
 
-    def __init__(self):
-        pass
-
     def backward(self, x):
         return invlogit(x, 0.0)
 

pymc3/tests/test_transforms.py

@@ -104,6 +104,16 @@ def test_log():
     close_to_logical(vals > 0, True, tol)
 
 
+def test_log_exp_m1():
+    check_transform_identity(tr.log_exp_m1, Rplusbig)
+    check_jacobian_det(tr.log_exp_m1, Rplusbig, elemwise=True)
+    check_jacobian_det(tr.log_exp_m1, Vector(Rplusbig, 2),
+                       tt.dvector, [0, 0], elemwise=True)
+
+    vals = get_values(tr.log_exp_m1)
+    close_to_logical(vals > 0, True, tol)
+
+
 def test_logodds():
     check_transform_identity(tr.logodds, Unit)
     check_jacobian_det(tr.logodds, Unit, elemwise=True)