Item response theory aims to estimate respondent's latent skills from their responses in tests composed of items with different levels of difficulty. Several models of item response theory have been proposed for different types of tasks, such as binary or probabilistic responses, response time, multiple responses, among others. In this paper, we propose a new version of $\beta^3$-IRT, called $\beta^{4}$-IRT, which uses the gradient descent method to estimate the model parameters. In $\beta^3$-IRT, abilities and difficulties are bounded, thus we employ link functions in order to turn $\beta^{4}$-IRT into an unconstrained gradient descent process. The original $\beta^3$-IRT had a symmetry problem, meaning that, if an item was initialised with a discrimination value with the wrong sign, e.g. negative when the actual discrimination should be positive, the fitting process could be unable to recover the correct discrimination and difficulty values for the item. In order to tackle this limitation, we modelled the discrimination parameter as the product of two new parameters, one corresponding to the sign and the second associated to the magnitude. We also proposed sensible priors for all parameters. We performed experiments to compare $\beta^{4}$-IRT and $\beta^3$-IRT regarding parameter recovery and our new version outperformed the original $\beta^3$-IRT. Finally, we made $\beta^{4}$-IRT publicly available as a Python package, along with the implementation of $\beta^3$-IRT used in our experiments.

翻译：项目反应理论旨在根据被试者对由不同难度项目组成的测验所作出的反应，来估计其潜在能力。针对不同类型的任务，如二元或概率型反应、反应时间、多项反应等，研究者已提出了多种项目反应理论模型。本文提出了一种新版本的$β^3$-IRT模型，即$β^{4}$-IRT，它采用梯度下降法来估计模型参数。在$β^3$-IRT中，能力和难度是有界的，因此我们使用链接函数将$β^{4}$-IRT转化为无约束的梯度下降过程。原始的$β^3$-IRT存在对称性问题，即若某个项目的区分度参数初始值符号错误（例如，实际区分度应为正时初始化为负），拟合过程可能无法正确恢复该项目的区分度和难度值。为解决这一局限，我们将区分度参数建模为两个新参数的乘积：一个对应符号，另一个对应大小。我们还为所有参数提出了合理的先验分布。我们通过实验比较了$β^{4}$-IRT和$β^3$-IRT在参数恢复方面的表现，新版本的性能优于原始的$β^3$-IRT。最后，我们将$β^{4}$-IRT作为Python包公开发布，同时包含实验中使用的$β^3$-IRT实现。