Efficient moment estimation with extremely small sample size via bayesian inference for analog/mixed-signal validation

Abstract

A critical problem in pre-Silicon and post-Silicon validation of analog/mixed-signal circuits is to estimate the distribution of circuit performances, from which the probability of failure and parametric yield can be estimated at all circuit configurations and corners. With extremely small sample size, traditional estimators are only capable of achieving a very low confidence level, leading to either over-validation or under-validation. In this paper, we propose a multipopulation moment estimation method that significantly improves estimation accuracy under small sample size. In fact, the proposed estimator is theoretically guaranteed to outperform usual moment estimators. The key idea is to exploit the fact that simulation and measurement data collected under different circuit configurations and corners can be correlated, and are conditionally independent. We exploit such correlation among different populations by employing a Bayesian framework, i.e., by learning a prior distribution and applying maximum a posteriori estimation using the prior. We apply the proposed method to several datasets including post-silicon measurements of a commercial highspeed I/O link, and demonstrate an average error reduction of up to 2×, which can be equivalently translated to significant reduction of validation time and cost.

BMF-Bd

“…An efficient BMF method has been proposed in [14] to improve the estimation accuracy of mean and variance with extremely small sample size. It relies on the assumption that the simulation and measurement data collected for different populations (i.e., different circuit configurations and/or corners) are strongly correlated.…”

“…The correlation information can thus be exploited by a Bayesian framework to improve the estimation accuracy. Although the authors of [14] mainly focus on the moment estimation problem for multiple populations, the proposed BMF method can be generally applied to other cases where mean and variance are accurately estimated by combining the prior information from an early stage with very few random samples at the late stage.…”

“…In practice, the variability of a performance of interest (say, x) can often be approximated as a Gaussian distribution [14]. As a result, the performance distribution is fully specified by its mean μ and variance σ 2 :…”

“…In this sub-section, we take the problem of mean estimation as an example to illustrate the BMF method proposed in [14]. Traditionally, the mean value μ in (1) is often estimated by the following estimator based on sample mean:…”

“…of circuit performance for both pre-silicon verification [12]- [13] and post-silicon validation [14]- [15]. BMF attempts to borrow the simulation and/or measurement data from an early stage to accurately estimate the statistics at the late stage with very few late-stage samples.…”