A point process on a space is a random bag of elements of that space. In this paper we explore programming with point processes in a monadic style. To this end we identify point processes on a space X with probability measures of bags of elements in X. We describe this view of point processes using the composition of the Giry and bag monads on the category of measurable spaces and functions and prove that this composition also forms a monad using a distributive law for monads. Finally, we define a morphism from a point process to its intensity measure, and show that this is a monad morphism. A special case of this monad morphism gives us Wald's Lemma, an identity used to calculate the expected value of the sum of a random number of random variables. Using our monad we define a range of point processes and point process operations and compositionally compute their corresponding intensity measures using the monad morphism.

中文翻译：

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概率点过程的Monad

空间上的点过程是该空间中元素的随机袋。在本文中，我们以单子形式探索点过程编程。为此，我们用空间中X的元素袋的概率度量来确定空间X上的点过程。我们使用Giry和bag monads在可测空间和函数类别上的构成来描述这种点过程的观点，并证明这种构成也使用分配法则为monad形成monad。最后，我们定义了一个从点过程到强度度量的态射，并证明这是一个单子态。这种单子态的特殊情况为我们提供了Wald引理，即用于计算随机数随机变量总和的期望值的标识。