高效序列Gibbs采样法：大规模认知诊断模型

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"这篇论文提出了一种用于处理大量属性的认知诊断模型的Sequential Gibbs采样算法，解决了当属性数量（K）较大时，传统Markov Chain Monte Carlo (MCMC)算法计算复杂度过高的问题。通过将时间复杂度从O(2^K)降低到O(K)，该方法显著提高了效率。在DINA模型和GDINA模型上进行了实验，并在仿真数据、分数减法数据集以及TIMMS 2007实际数据集上验证了其性能。对于K=15，I=2000，J=30的参数设置，模型在3000次迭代中仅需170秒，显示了高效性。关键词包括Gibbs采样、GDINA、参数估计和认知诊断。" 认知诊断模型(Cognitive Diagnosis Models, CDMs)是统计学中一种强大的工具，它们能够提供有关干预和学习的深入信息。在教育和心理学等领域，CDMs被广泛应用于评估个体的知识掌握程度和能力结构。其中，DINA模型（Diagnostic Model for Multiple Attributes）和GDINA（Generalized DINA）模型是两种常用模型，分别代表了不同的假设和复杂度。传统的MCMC算法是估计和推断CDMs的常见方法，但在属性数量大时，由于需要计算每个属性配置的条件分布，其时间复杂度高，导致计算效率低下。为了克服这个问题，作者提出了一个序列Gibbs采样方法。这种新方法通过减少计算量，使得每次采样属性配置仅需O(K)的计算，极大地提升了效率。在模拟数据和真实数据实验中，Sequential Gibbs采样展示出了良好的有限样本性能，并且在处理大规模属性集时比现有方法具有优势。例如，当K=15，I=2000，J=30的条件下，模型在3000次迭代中只需170秒，这表明新算法在处理大问题时的计算速度显著快于传统方法。这篇论文提出的Sequential Gibbs采样算法为解决CDMs的大规模属性估计问题提供了一个高效解决方案，对于需要快速处理大量数据的领域，如大规模在线学习评估，具有重要的实践意义。

2.2 Priors of Measurement Models’ Parameters

The population proportion parameter π includes t he saturated information about the

attribute proﬁle distribution in CDMs. The Dirichlet distribution is commonly used as

a conjugated prio r for π, such as Culpepper (2015), Culpepper and Hudson (2018) and

Zhan et al. (2019). The speciﬁc form of the Dirichlet prior for π is

p(π) = p(π

, · · · , π

) = Dirichlet(δ

, · · · , δ

Let the notation δ = (δ, · · · , δ

)

′

represent a C-dimensional hyper-par ameter vector of π,

and we denote the prior p(π) by Dir ichlet(δ). The Dirichlet prior implies the generation

of attribute proﬁles, i.e., δ determines π and π determines α.

In diﬀerent CDMs, item parameters will be presented in the diﬀerent f orms: g

and

are item parameters in the DINA model, respectively, and λ

is the item parameter

in the GDINA model. For the DINA model, independent Beta distributions, Beta(a

, b

)

and Beta(a

, b

), are often used as the prio r s for guessing and slipping parameters, respec-

tively. We may also constraint 0 ≤ g

< 1 − s

≤ 1, to ensure the mo del identiﬁability

(Junker and Sijtsma, 2001; Chen et al., 2018; Xu and Zhang , 2016; Gu and Xu, 2019).

For the GDINA model, the item parameters λ are r egr ession coeﬃcients. The normal

distributions are often taken as priors for the item parameters (e.g., Zhan et a l., 2019).

Speciﬁcally, two types of pr iors are often chosen for item parameters, the one is a multi-

variate normal distribution, λ

∼ N(µ

, Σ

), as a general choice; the other is a truncated

multivariate normal distribution, λ

∼ N(µ

, Σ

{λ

∈T }

, which is used to ensure cer-

tain monotonicity assumption of the item resp onse function. Particularly, assuming the

dimension of λ

is M, T = { T

, · · · , T

} and each set T

represents some pr e-speciﬁed

constraint of the m-th element of λ

. For instance, we may r estrict the main eﬀect terms

in λ

to be positive to ensure the monotonicity assumption. For the indicator function

{λ

∈T }

, I

{λ

∈T }

= 1 if λ

∈ T

for m = 1, · · · , M and I

{λ

∈T }

= 0 otherwise.

As discussed in the introduction, for a large K, the existing MCMC algorithms with

Dirichlet prior for π often suﬀer from the increasing computationa l cost of sampling

each latent attribute proﬁle α

from its conditiona l distribution, which is a categorical

distribution with 2

diﬀerent categories. Therefore, it needs to evaluate 2

posterior

probabilities of each possible proﬁle candidate to sample each α

, and the corresponding

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