  The requirement-driven performance evaluation of a black-box cyber-physical
system (CPS) that utilizes machine learning methods has proven to be an
effective way to assess the quality of the CPS. However, the distributional
evaluation of the performance has been poorly considered. Although many
uncertainty estimation methods have been advocated, they have not successfully
estimated highly complex performance distributions under small data. In this
paper, we propose a method to distributionally evaluate the performance under
complex requirements using small input-trajectory data. To handle the unknown
complex probability distributions under small data, we discretize the
corresponding performance measure, yielding a discrete random process over an
input region. Then, we propose a semiparametric Bayesian model of the discrete
process based on a Dirichlet random field whose parameter function is
represented by multiple logistic Gaussian processes (LGPs). The Dirichlet
posterior parameter function is estimated through the LGP posteriors in a
reasonable and conservative fashion. We show that the proposed Bayesian model
converges to the true discrete random process as the number of data becomes
large enough. We also empirically demonstrate the effectiveness of the proposed
method by simulation.
