Developing Bayesian-based Confidence Bounds for Non-identically Distributed Observations using the Lyapunov Condition

The purpose of this paper is to establish a direct method for assessing the confidence in the detection and identification probabilities for segmented observations that are not identically distributed across assigned segments within a region. This paper arrives at easily computable confidence intervals by showing through mathematical analysis that: I. The probability of successful detection within each test segment can be characterized by a Beta distribution; II. The distribution of a weighted sum of independent but non-identically distributed sample means is asymptotically Normally distributed by the Lyapunov variant of the Central Limit Theorem, i.e., the approximation improves as the number of samples increases; III. Given that the distribution of the sample means convergences to a Normal distribution, the confidence intervals about the observed sample means for both the detection and identification probabilities can be determined in closed form for multiple target types. The motivation for this approach is the need to determine the exceedance probabilities to support a Systems Acceptance Test based on collected data.