How to calculate log posterior of a GP over a trace in pymc3

Questions
2

Yes you can do it this way to save the logp into the trace, otherwise you can evaluate it outside of the model by creating/accessing the right function.

Some related discussion see Frequently Asked Questions

So you can do:

with model:
     logp_y1_ = pm.Deterministic('logp', y1_.logpt)

then do sample.

Or you can sample first, and then evaluate the logp of y1_ (notice that I also move the conditional outside, so the trace is smaller):

with pm.Model() as model:
    l = pm.HalfNormal('l',5.)
    cov_func = pm.gp.cov.ExpQuad(1, ls=l)
    gp = pm.gp.Marginal(cov_func=cov_func)
    y0_ = gp.marginal_likelihood('y0',X0,y0,0.1)
    trace = pm.sample(100)

with model:
    y1_ = gp.conditional('y1',X1,given={'X':X0,'y':y0,'noise':0.1})
# evaluate logp on one input from the posterior:
point = trace[0] # get a point from trace
point['y1'] = y1
y1_.logp(point)