在序贯推断中将鞅扩散嵌入为二元后验概率
Embedding martingale diffusions as binary posteriors in sequential inference
AI总结：
研究将鞅扩散嵌入二元后验概率的问题，通过显式二元序贯实验实现，给出了包括吸收布朗运动等在内的有界鞅扩散的序贯推断解释，揭示了相关条件及不同情况下后验性质。
AI中文摘要：
后验概率是有界鞅，但仅这一事实并不能确定产生它们的实验。我们表明，在[0,1]上的每个时间齐次鞅扩散，其内部具有严格正的\(C^1\)波动率且端点为吸收型，即我们所称的自主赢鞅，可实现为通过标量扩散观测的显式二元序贯实验的贝叶斯后验。反之，在一般二元扩散实验中，当两个假设下信号漂移的差异满足里卡蒂方程时，当前时间和观测对于隐藏状态是联合充分的。当似然比由与零动态的正时空调和函数相关的杜布\(h\)变换生成时，情况也是如此。在这类情况中，当该函数是空间调和时，后验是自主鞅。在非调和情况下，自主性迫使漂移差异为常数，从二元测试中恢复经典的两状态 Shiryaev - Wonham 滤波器。这些结果为包括吸收布朗运动、赖特 - 费希尔扩散和“最刺激游戏”的奥尔德斯鞅在内的一大类有界鞅扩散给出了具体的序贯推断解释。
英文摘要：
Posterior probabilities are bounded martingales, but this fact alone does not identify the experiment that generates them. We show that every time-homogeneous martingale diffusion on $[0,1]$ with strictly positive $C^1$ volatility on the interior and absorbing endpoints, which we call an autonomous win-martingale, can be realized as the Bayesian posterior of an explicit binary sequential experiment observed through a scalar diffusion. Conversely, in a general binary diffusion experiment, the current time and observation are jointly sufficient for the hidden state precisely when the difference between the signal drifts under the two hypotheses satisfies a Riccati equation. Equivalently, this holds when the likelihood ratio is generated by a Doob $h$-transform associated with a positive space-time harmonic function for the null dynamics. Within this class, when that function is spatially harmonic, the posterior is an autonomous martingale. Outside this harmonic case, autonomy forces the drift difference to be constant, recovering the classical two-state Shiryaev-Wonham filter from binary testing. These results give concrete sequential inference interpretations to a broad class of bounded martingale diffusions, including absorbed Brownian motion, the Wright-Fisher diffusion, and the Aldous martingale of the "most exciting game."
