The maximum mean discrepancy (MMD) between two probability measures P
 and Q is a metric that is zero if and only if all moments of the two measures
 are equal, making it an appealing statistic for two-sample tests. Given i.i.d. samples
 from P and Q, Gretton et al. (2012) show that we can construct an unbiased
 estimator for the square of the MMD between the two distributions. If P is a
 distribution of interest and Q is the distribution implied by a generative neural
 network with stochastic inputs, we can use this estimator to train our neural network.
 However, in practice we do not always have i.i.d. samples from our target
 of interest. Data sets often exhibit biases—for example, under-representation of
 certain demographics—and if we ignore this fact our machine learning algorithms
 will propagate these biases. Alternatively, it may be useful to assume our data has
 been gathered via a biased sample selection mechanism in order to manipulate
 properties of the estimating distribution Q.
 In this paper, we construct an estimator for the MMD between P and Q when we
 only have access to P via some biased sample selection mechanism, and suggest
 methods for estimating this sample selection mechanism when it is not already
 known. We show that this estimator can be used to train generative neural networks
 on a biased data sample, to give a simulator that reverses the effect of that
 bias.
We propose an estimator for the maximum mean discrepancy, appropriate when a target distribution is only accessible via a biased sample selection procedure, and show that it can be used in a generative network to correct for this bias. Proposes an importance-weighted estimator of the MMD to estimate the MMD between distributions based on samples biased according to a known or estimated unknown scheme. The authors address the problem of sample selection bias in MMD-GANs and propose an estimate of the MMD between two distributions using weighted maximum mean discrepancy. This paper presents a modification of the objective used to train generative networks with an MMD adversary 