  The fully marginalized likelihood, or Bayesian evidence, is of great
importance in probabilistic data analysis, because it is involved in
calculating the posterior probability of a model or re-weighting a mixture of
models conditioned on data. It is, however, extremely challenging to compute.
This paper presents a geometric-path Monte Carlo method, inspired by
multi-canonical Monte Carlo to evaluate the fully marginalized likelihood. We
show that the algorithm is very fast and easy to implement and produces a
justified uncertainty estimate on the fully marginalized likelihood. The
algorithm performs efficiently on a trial problem and multi-companion model
fitting for radial velocity data. For the trial problem, the algorithm returns
the correct fully marginalized likelihood, and the estimated uncertainty is
also consistent with the standard deviation of results from multiple runs. We
apply the algorithm to the problem of fitting radial velocity data from HIP
88048 ($\nu$ Oph) and Gliese 581. We evaluate the fully marginalized likelihood
of 1, 2, 3, and 4-companion models given data from HIP 88048 and various
choices of prior distributions. We consider prior distributions with three
different minimum radial velocity amplitude $K_{\mathrm{min}}$. Under all three
priors, the 2-companion model has the largest marginalized likelihood, but the
detailed values depend strongly on $K_{\mathrm{min}}$. We also evaluate the
fully marginalized likelihood of 3, 4, 5, and 6-planet model given data from
Gliese 581 and find that the fully marginalized likelihood of the 5-planet
model is too close to that of the 6-planet model for us to confidently decide
between them.
