Limited labeling budget severely impedes data-driven research, such as medical analysis, remote sensing and population census, and active inference is a solution to this problem. Prior works utilizing independent sampling have achieved improvements over uniform sampling, but its insufficient usage of available information undermines its statistical efficiency. In this paper, we propose balanced active inference, a novel algorithm that incorporates balanced constraints based on model uncertainty utilizing the cube method for label selection. Under regularity conditions, we establish its asymptotic properties and also prove that the statistical efficiency of the proposed algorithm is higher than its alternatives. Various numerical experiments, including regression and classification in both synthetic setups and real data analysis, demonstrate that the proposed algorithm outperforms its alternatives while guaranteeing nominal coverage.

Abstract Improving energy efficiency by adjusting the structure of energy consumption types is of great significance for reducing carbon emissions in the short term. The present paper constructs new data envelopment analysis models for evaluating energy utilization under different structural conditions and calculating potential emissions reductions. We conducted empirical research on 30 provinces in China from 2003 to 2019—a time frame that coincides with the instituting of China's “coal‐to‐gas” program. Our results show that technological progress is the main way for China to reduce carbon emissions and that it is possible to reduce the total amount of carbon emissions by 35%. Additionally, optimizing the energy consumption structure following the coal‐to‐gas program guidelines could reduce the country's carbon emissions by a further 25%. Finally, this paper provides specific policy recommendations based on the efficiency analysis results to guide each province in reducing carbon emissions under the conditions of energy demand growth.

Abstract Environmental regulation has been recognized as an important way to directly improve environmental performance or indirectly impact environmental performance through increasing environmental innovation. The present paper constructs an energy and carbon emission total factor productivity index using the Malmquist–Luenberger productivity index and data envelopment analysis. The proposed technique is used to measure the environmental performance of 30 Chinese provinces during the period 2010–2015. The energy and carbon emission total factor productivity measure is divided into a pure technical efficiency index and a technical progress index to provide detailed environmental performance information. Then an ordinary least squares model is adopted to analyse the impact of environmental regulation on environmental innovation operation and environmental performance by hypothesis testing. The empirical results show that environmental regulation and environmental innovation have positive direct effects on environmental performance and that market‐based environmental regulation has a positive indirect effect on environmental performance by increasing environmental innovation.