A new framework of multistage parametric inference

Coalescence of low-viscosity fluids in air

“…Here, by "controlled", we mean that the coverage probabilities can be adjusted to be above any desirable level by making ζ > 0 sufficiently small. To make the coverage probability of a sequential random interval controllable by ζ, we propose to use a sequence of confidence intervals whose coverage probabilities can be controlled by ζ, referred to as controlling confidence sequence, to determine a stopping rule such that the sequential random interval must include the controlling confidence sequence at the termination of sampling process (see, e.g., Section 3 of the fifth version of our paper [18] published in arXiv on April 7, 2009, our SPIE paper [20] published in April 2010, and our earlier versions of this paper from September 2008 to present). We call such a methodology of using confidence sequences to define stopping rules to control the coverage probabilities of the associated sequential random intervals as inclusion principle.…”

“…For many problems, if interval [a, b] is narrow enough, then, condition (22) can be satisfied and the upper and lower bounds of Pr{L ( θ, n) ≥ θ or U ( θ, n) ≤ θ | θ} in (20) and (21) can be used to determine whether Pr{L (…”

“…This suggests an alternative approach for constructing sequential random intervals to guarantee prescribed confidence level for any θ ∈ [θ, θ], where [θ, θ] is a subset of parameter space Θ. The basis idea is as follows: (i) Construct sampling scheme such that the probabilities Pr{θ ≤ L ( θ, n) | θ} and Pr{θ ≥ (20) and (21) can be used to determine…”

Scalable Ensemble Learning by Adaptive Sampling

“…Sample size could be determined a priori as in conventional batch sampling, or it could be dynamically determined as in adaptive, sequential sampling. As recent studies [4], [13] pointed out, there are situations in which using Chernoff/Hoeffding bounds in "static" (non-adaptive) sampling would require a sample size that is unnecessarily large. In contrast, adaptive sampling decides whether it has seen sufficient samples based some criterion related to the samples seen so far.…”

“…In a recent work [3] closely related to [4], a new adaptive sequential sampling method was developed that can handle cases of controlling absolute error and relative error. Empirical results were shown in [3] indicating that the new sampling method uses significantly lower sample size compared to the method in [13].…”