We make two closely related theoretical contributions to the use of importance sampling schemes. First, for independent sampling, we prove that the minimax optimal trial distribution coincides with the target if and only if the target distribution has no atom with probability greater than $1/2$, where "minimax" means that the worst-case asymptotic variance of the self-normalized importance sampling estimator is minimized. When a large atom exists, it should be downweighted by the trial distribution. A similar phenomenon holds for a continuous target distribution concentrated on a small set. Second, we argue that it is often advantageous to run the Metropolis--Hastings algorithm with a tempered stationary distribution, $π(x)^β$, and correct for the bias by importance weighting. The dynamics of this "importance-tempered" sampling scheme can be described by a continuous-time Markov chain. We prove that for one-dimensional targets with polynomial tails, $π(x) \propto (1 + |x|)^{-γ}$, this chain is uniformly ergodic if and only if $1/γ< β< (γ- 2)/γ$. These results suggest that for target distributions with light or polynomial tails of order $γ> 3$, importance tempering can improve the precision of time-average estimators and essentially eliminate the need for burn-in.

The interplay between cognition and gaming, notably through educational games enhancing cognitive skills, has garnered significant attention in recent years. This research introduces the CogSimulator, a novel algorithm for simulating user cognition in small-group settings with minimal data, as the educational game Wordle exemplifies. The CogSimulator employs Wasserstein-1 distance and coordinates search optimization for hyperparameter tuning, enabling precise few-shot predictions in new game scenarios. Comparative experiments with the Wordle dataset illustrate that our model surpasses most conventional machine learning models in mean Wasserstein-1 distance, mean squared error, and mean accuracy, showcasing its efficacy in cognitive enhancement through tailored game design.

The concept of free energy has its origins in 19th century thermodynamics, but has recently found its way into the behavioral and neural sciences, where it has been promoted for its wide applicability and has even been suggested as a fundamental principle of understanding intelligent behavior and brain function. We argue that there are essentially two different notions of free energy in current models of intelligent agency, that can both be considered as applications of Bayesian inference to the problem of action selection: one that appears when trading off accuracy and uncertainty based on a general maximum entropy principle, and one that formulates action selection in terms of minimizing an error measure that quantifies deviations of beliefs and policies from given reference models. The first approach provides a normative rule for action selection in the face of model uncertainty or when information-processing capabilities are limited. The second approach directly aims to formulate the action selection problem as an inference problem in the context of Bayesian brain theories, also known as Active Inference in the literature. We elucidate the main ideas and discuss critical technical and conceptual issues revolving around these two notions of free energy that both claim to apply at all levels of decision-making, from the high-level deliberation of reasoning down to the low-level information-processing of perception.