论文标题
贝叶斯顺序检测和估计多个假设
Bayesian Sequential Joint Detection and Estimation under Multiple Hypotheses
论文作者
论文摘要
我们考虑共同检验多个假设并估算基础分布的随机参数的问题。在对基础随机过程的轻度假设下的顺序设置中研究了这个问题。最佳方法最小化了预期的样品数量，同时确保平均检测/估计误差不超过一定水平。将约束问题转换为不受约束的问题后，我们通过非线性钟形方程来表征通用解决方案，该方程是由一组成本系数进行参数化的。得出了相对于系数的成本函数衍生物与顺序过程的检测/估计误差之间的牢固联系。基于此基本属性，我们进一步表明，对于适当选择的成本系数，约束和不受约束的问题的解决方案重合。我们提出了找到最佳系数的两种方法。对于第一种方法，最终的优化问题将转换为线性程序，而第二种方法则以预测的梯度上升来解决它。为了说明理论结果，我们考虑了两个问题，这些问题是数值设计的。使用Monte Carlo模拟，验证了数值结果与该理论一致。
We consider the problem of jointly testing multiple hypotheses and estimating a random parameter of the underlying distribution. This problem is investigated in a sequential setup under mild assumptions on the underlying random process. The optimal method minimizes the expected number of samples while ensuring that the average detection/estimation errors do not exceed a certain level. After converting the constrained problem to an unconstrained one, we characterize the general solution by a non-linear Bellman equation, which is parametrized by a set of cost coefficients. A strong connection between the derivatives of the cost function with respect to the coefficients and the detection/estimation errors of the sequential procedure is derived. Based on this fundamental property, we further show that for suitably chosen cost coefficients the solutions of the constrained and the unconstrained problem coincide. We present two approaches to finding the optimal coefficients. For the first approach, the final optimization problem is converted into a linear program, whereas the second approach solves it with a projected gradient ascent. To illustrate the theoretical results, we consider two problems for which the optimal schemes are designed numerically. Using Monte Carlo simulations, it is validated that the numerical results agree with the theory.
