本研究將文獻上潛在空間試題反應理論模型中的連續潛在空間改良至離散潛在空間,此方法可以更精準地估計受試者潛在特徵和試題難度參數,並降低有效參數數量。離散潛在空間仍然可以呈現受試者與試題之間的相對關係。本研究採用貝氏估計,並以 spike-and-slab 先驗分布 (Ishwaran & Rao 2005; Rockova & George 2018) 進行模型選取,並使用吉布斯抽樣與 Metropolis-Hastings 演算法等計算技巧,完成模型選取以及參數估計。透過模擬方法呈現參數估計之均方根誤差以及受試者與試題在潛在空間中分布。最後本文將提出之方法應用於一筆實際資料並與連續潛在空間之試題反應理論模型比較。

This study extends the continuous latent space employed in existing latent-space item response theory models to a discrete latent space. The proposed approach enables more accurate estimation of respondents’ latent traits and item difficulty parameters while reducing the effective number of parameters. Despite its discrete structure, the latent space can still represent the relative relationships between respondents and items.
This study adopts a Bayesian estimation framework and employs a spike-and-slab prior distribution (Ishwaran & Rao, 2005; Rockova & George, 2018) for model selection. Computational techniques, including Gibbs sampling and the Metropolis–Hastings algorithm, are used to perform model selection and parameter estimation. A simulation study is conducted to evaluate the root mean squared errors of the parameter estimates.
Finally, the proposed method is applied to an empirical dataset and compared with an item response theory model based on a continuous latent space.

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