In this paper, we are concerned with estimating the joint probability of
random variables $X$ and $Y$, given $N$ independent observation blocks
$(\boldsymbol{x}^i,\boldsymbol{y}^i)$, $i=1,\ldots,N$, each of $M$ samples
$(\boldsymbol{x}^i,\boldsymbol{y}^i) = \bigl((x^i_j, y^i_{\sigma^i(j)})
\bigr)_{j=1}^M$, where $\sigma^i$ denotes an unknown permutation of i.i.d.
sampled pairs $(x^i_j,y_j^i)$, $j=1,\ldots,M$. This means that the internal
ordering of the $M$ samples within an observation block is not known. We derive
a maximum-likelihood inference functional, propose a computationally tractable
approximation and analyze their properties. In particular, we prove a
$\Gamma$-convergence result showing that we can recover the true density from
empirical approximations as the number $N$ of blocks goes to infinity. Using
entropic optimal transport kernels, we model a class of hypothesis spaces of
density functions over which the inference functional can be minimized. This
hypothesis class is particularly suited for approximate inference of transfer
operators from data. We solve the resulting discrete minimization problem by a
modification of the EMML algorithm to take addional transition probability
constraints into account and prove the convergence of this algorithm.
Proof-of-concept examples demonstrate the potential of our method.

本文关注的是给定 N 个独立观测块的随机变量 X 和 Y 的联合概率的估计问题，我们推导了极大似然推断的函数形式，提出了一种可计算的逼近方法，并分析了它们的性质。我们证明了一个 Γ- 收敛结果，表明随着观测块的数量 N 趋于无穷大，我们能够从经验逼近中恢复真实的概率密度。通过熵最优输运核，我们建模了一类假设空间，通过最小化推断函数在该假设类上的值，可以近似推断数据中的转移算子。我们通过修改 EMML 算法以考虑额外的转移概率约束来解决所得到的离散最小化问题，并证明了该算法的收敛性。概念验证示例展示了我们方法的潜力。