first stab at integrated prior; works somewhat

[maxbiostat]

Please do not merge yet

The idea

Place a proper prior on the (constant) population size Ne and then integrate it out of the coalescent likelihood.
(i) Employing an inverse-Gamma prior on Ne, with parameters alpha and beta one gets a nice closed-form density. This is clumsily implemented in src/dr/evomodel/coalescent/IGCoalescentLikelihood.java.
(ii) Later, one can integrate over beta from 0 to u to potentially get a prior that does not need user input. This has not yet been implemented.

Tests

Tests compare the results of using the integrated prior with alpha = 2.0and beta = 10.0 - giving a prior expectation of E[Ne] = 10 - and the same prior on Ne but sampling its distribution using a standard ScaleOperator (full).
All tests were run with beast -seed 666 -overwrite for 10M iterations, sampling every 100. Only one replicate was run.
Results are shown for tree height and tree length (sum of branch lenghts).

• simulate from the prior with 5 contemporaneous taxa:
  note: Root height should be 2*10*(4/5) = 16.

Quantity	Full	Marginalised
Wall clock time (minutes)	2.6	1.1
Root height mean (sd)	16.6 (27.4)	16.6 (27.4)
Tree length mean (sd)	42.6 (69.6)	42.4 (60.6)
Root height ESS	3342	3807
Tree length ESS	3198	4027

• simulate from the prior with 50 contemporaneous taxa:
  note: root height should be round about 2*10*49/50 = 19.6

Quantity	Full	Marginalised
Wall clock time (minutes)	3.17	3.32
Root height mean (sd)	18.6 (32.2)	19.5 (29.8)
Tree length mean (sd)	85 (130)	89 (69)
Root height ESS	858	3165
Tree length ESS	686	2351

• simulate from the prior with 17 heterochronous taxa:

Quantity	Full	Marginalised
Wall clock time (minutes)	2.6	2.7
Root height mean (sd)	52.2 (46.5)	51.1 (27.2)
Tree length mean (sd)	130 (157)	167 (64)
Root height ESS	509	2206
Tree length ESS	413	2411

• sample from the posterior with 17 heterochronous taxa:

Quantity	Full	Marginalised
Wall clock time (minutes)	5.8	6.0
Root height mean (sd)	68.6 (9.7)	57.3 (7.9)
Tree length mean (sd)	258 (37)	212 (20)
Root height ESS	2038	472
Tree length ESS	1793	398

• sample from a simplified posterior (simpler model) with 17 heterochronous taxa:

Quantity	Full	Marginalised
Wall clock time (minutes)	3.5	3.5
Root height mean (sd)	67.4 (5.8)	61.2 (4.5)
Tree length mean (sd)	263 (28)	231 (21)
Root height ESS	3892	4132
Tree length ESS	3242	3278

Notes detailing the calculations are available upon request.

all_xmls.zip