Add test case for Loopy ARG

[hyanwong]

We have the simplest possible loopy graph in

tsdate/tests/utility_functions.py

Line 676 in 7282a30

def loopy_tree():

We should use this in the test_accuracy.py file somehow, to check that iteration in the Expectation Propagation algorithm correctly accounts for the probabilities passed around the graph. I think there should be a semi-principled way of thing this (akin to checking the total functional value around the tree, e.g. in

tsdate/tests/test_functions.py

Line 1270 in 7282a30

class TestTotalFunctionalValueTree:

). Can you think of any way to test this @nspope: i.e. that's not a test of the accuracy of time inference, but of the end point of the iteration? E.g. in the simplest case, can we find a predicted set of values we should be converging on?

[nspope]

Hmm, some general thoughts:

• The only case where we can compute the posterior exactly is a three tip tree (two free nodes); but in this case the tests reduce to the "TestPosteriorMomentMatching" class.
• Because of the time-ordered traversal, EP for a single tree should converge in a single iteration; we don't test this currently
• If we add a lot of mutations to a single tree, then the EP estimates should get consistent rather quickly; e.g. the posterior should concentrate around the true mean. This would be good to test (also for numerical problems).
• We can test that the approximations converge to something -- the gamma parameters stabilizing after a few iterations. We can just measure the norm of the difference in sufficient statistics between two consecutive iterations or something similar.