A new model to predict weak-lensing peak counts II. Parameter constraint
  strategies
  Peak counts have been shown to be an excellent tool to extract the
non-Gaussian part of the weak lensing signal. Recently, we developped a fast
stochastic forward model to predict weak-lensing peak counts. Our model is able
to reconstruct the underlying distribution of observables for analyses. In this
work, we explore and compare various strategies for constraining parameter
using our model, focusing on the matter density $\Omega_\mathrm{m}$ and the
density fluctuation amplitude $\sigma_8$. First, we examine the impact from the
cosmological dependency of covariances (CDC). Second, we perform the analysis
with the copula likelihood, a technique which makes a weaker assumption
compared to the Gaussian likelihood. Third, direct, non-analytic parameter
estimations are applied using the full information of the distribution. Fourth,
we obtain constraints with approximate Bayesian computation (ABC), an
efficient, robust, and likelihood-free algorithm based on accept-reject
sampling. We find that neglecting the CDC effect enlarges parameter contours by
22%, and that the covariance-varying copula likelihood is a very good
approximation to the true likelihood. The direct techniques work well in spite
of noisier contours. Concerning ABC, the iterative process converges quickly to
a posterior distribution that is in an excellent agreement with results from
our other analyses. The time cost for ABC is reduced by two orders of
magnitude. The stochastic nature of our weak-lensing peak count model allows us
to use various techniques that approach the true underlying probability
distribution of observables, without making simplifying assumptions. Our work
can be generalized to other observables where forward simulations provide
samples of the underlying distribution.
