2.1 Model setup

The DeepCDM framework is developed to address the need for diagnostic modeling at multiple granularities. It is defined using the terminology of probabilistic graphical models (Koller & Friedman, Reference Koller and Friedman2009; Wainwright & Jordan, Reference Wainwright and Jordan2008), particularly directed graphical models. These models employ graphs to compactly represent the joint distribution of high-dimensional random variables, where nodes correspond to variables and edges encode their direct probabilistic relationships.

We first review the definition of a directed acyclic graph (DAG), also referred to as a Bayesian network (Pearl, Reference Pearl1988). In a DAG, every edge has a direction, and no directed cycles are allowed. Consider M random variables, $X_1, \ldots , X_M$ , which correspond to M nodes in the graph. If a directed edge goes from $X_\ell $ to $X_m$ , we say that $X_\ell $ is a parent of $X_m$ , and $X_m$ is a child of $X_\ell $ . Let ${\mathrm{pa}}(m) \subseteq \{1, \dots , M\}$ denote the index set of all parents of $X_m$ . Define $\mathbb {P}(X_m \mid X_{{\mathrm{pa}}(m)})$ as the conditional distribution of $X_m$ given its parents $X_{{\mathrm{pa}}(m)}$ . Based on this DAG structure, the joint distribution of $X_1, \ldots , X_M$ factorizes as follows:

(1) $$ \begin{align} \mathbb{P}(X_1, \ldots, X_M) = \prod_{m=1}^{M} \mathbb{P}(X_m \mid X_{{\mathrm{pa}}(m)}). \end{align} $$

We now present the general DeepCDM formulation. For a DeepCDM with D latent layers, we denote the d-th latent layer as $\mathbf A^{(d)} = (A^{(d)}_1, \dots , A^{(d)}_{K_d})$ for each $d = 1, 2, \dots , D$ , where larger d correspond to deeper layers. In DeepCDMs, all edges are directed top-down and occur only between adjacent layers, defining a generative process from high-level latent variables to observed responses. Specifically, the bottom layer consists of the observed response variables for the J items, denoted as $\mathbf R = (R_1, \dots , R_J)$ . The first latent layer, right above the bottom layer, captures the most fine-grained latent attributes, represented as $\mathbf A^{(1)}$ . These are generated from the second latent layer $ \mathbf A^{(2)} $ , and the process continues recursively up to the deepest layer $ \mathbf A^{(D)} $ . Figure 1 gives an example of a DeepCDM with three latent layers ( $D=3$ ). Given the variables in the layer right above, the variables within each layer of the DeepCDM are conditionally independent. This structure intuitively models how more specific latent skills are successively derived from more general, higher-level latent “meta-skills.” A natural assumption, supported by the model’s identifiability conditions, is that deeper layers should consist of fewer latent variables, i.e., $K_1> K_2 > \dots > K_D$ (see Theorems 1, 2, and 3 in the Supplementary Material for detailed identifiability conditions).

Figure 1

A ladder-shaped three-latent-layer DeepCDM.

Note: Gray nodes are observed variables, and white nodes are latent ones. Multiple layers of binary latent variables $\mathbf A^{(1)}$ , $\mathbf A^{(2)}$ , and $\mathbf A^{(3)}$ successively generate the observed binary responses $\mathbf R$ . Binary matrices $\mathbf Q^{(1)}$ , $\mathbf Q^{(2)}$ , and $\mathbf Q^{(3)}$ encode the sparse connection patterns between adjacent layers in the graph.

In traditional CDMs with a single layer of K latent attributes, the $\mathbf Q$ -matrix (Tatsuoka, Reference Tatsuoka1983) is a fundamental component that specifies the relationship between items and the latent attributes. Specifically, $\mathbf Q = (q_{j,k})_{J \times K}$ , where $q_{j,k} = 1$ if the item j measures the latent attribute k, and $q_{j,k} = 0$ otherwise. Since the edges in a graphical model reflect direct statistical dependencies, $q_{j,k} = 1$ or $0$ also conveys whether the k-th latent node is a parent of the j-th observed node. Consequently, the $\mathbf Q$ -matrix encodes the bipartite graph structure between the observed and latent layers. Extending this idea to DeepCDMs, with D latent layers, requires D matrices, denoted as $\mathbf Q^{(1)}, \mathbf Q^{(2)}, \dots , \mathbf Q^{(D)}$ , to capture the dependence relationships between any two adjacent layers. Specifically, $\mathbf Q^{(1)} = \left (q^{(1)}_{j,k}\right )$ has size $J \times K_1$ , similar to the single $\mathbf Q$ -matrix in traditional CDM, and describes the graph between the observed data layer and the shallowest latent layer. While for $d = 2, \dots , D$ , the matrix $\mathbf Q^{(d)} = \left (q^{(d)}_{k,\ell }\right )$ has size $K_{d-1} \times K_d$ and represents the dependencies between latent variables in the $(d-1)$ th and dth latent layers. The entry $q^{(d)}_{k,\ell } = 1$ or $0$ indicates whether the latent variable $A^{(d)}_{\ell }$ is a parent of $A^{(d-1)}_k$ . In this article, we consider the challenging setting of exploratory DeepCDMs, where all $\mathbf Q$ -matrices are unknown and need to be estimated.

Based on the general definition of DAGs in (1) and the DeepCDM setup, the joint distribution of all variables, including the latent ones, is given by

(2) $$ \begin{align} \mathbb{P}(\mathbf R, \mathbf A^{(1)}, \dots, \mathbf A^{(D)}) &= \mathbb{P}(\mathbf R \mid \mathbf A^{(1)}, \mathbf Q^{(1)}) \cdot \prod_{d=2}^{D} \mathbb{P}(\mathbf A^{(d-1)} \mid \mathbf A^{(d)}, \mathbf Q^{(d)}) \cdot \mathbb{P}(\mathbf A^{(D)}), \end{align} $$

(3) $$ \begin{align} \text{where} \quad \mathbb{P}(\mathbf R = \boldsymbol r \mid \mathbf A^{(1)}, \mathbf Q^{(1)}) &= \prod_{j=1}^{J} \mathbb P^{\mathrm{CDM}}(R_j = r_j \mid \mathbf A^{(1)}, \mathbf Q^{(1)}), \quad \text{and} \end{align} $$

(4) $$ \begin{align} \mathbb{P}(\mathbf A^{(d-1)} = \boldsymbol \alpha^{(d-1)} \mid \mathbf A^{(d)}, \mathbf Q^{(d)}) &= \prod_{k=1}^{K_{d-1}} \mathbb P^{\mathrm{CDM}}(A^{(d-1)}_k = \alpha^{(d-1)}_k \mid \mathbf A^{(d)}, \mathbf Q^{(d)}), \end{align} $$

where $\boldsymbol r$ represents an observed response pattern and $\boldsymbol \alpha ^{(d-1)}$ represents a latent pattern for the $(d-1)$ th latent layer. The superscript “CDM” in the conditional distributions of (3) and (4) indicates that the conditional distribution within each layer of the generative process adheres to a CDM. By marginalizing out all latent layers $\mathbf A^{(1)}, \dots , \mathbf A^{(D)}$ in (2), we obtain the marginal distribution of the observed response vector $\mathbf R$ :

(5) $$ \begin{align} \mathbb{P}(\mathbf R = \boldsymbol r) &= \sum_{\boldsymbol \alpha^{(1)}} \cdots \sum_{\boldsymbol \alpha^{(D)}} \mathbb{P}(\mathbf R = \boldsymbol r, \mathbf A^{(1)} = \boldsymbol \alpha^{(1)}, \dots, \mathbf A^{(D)} = \boldsymbol \alpha^{(D)}). \end{align} $$

This work focuses on binary observed and latent variables, where $\boldsymbol r \in \{0,1\}^J$ and $\boldsymbol \alpha ^{(d)} \in \{0,1\}^{K_d}$ . Each observed variable reflects whether a response is correct or incorrect, while each latent variable indicates the presence or absence of a specific skill or higher-level attribute. Similar to traditional CDMs, the latent variables $\mathbf A^{(D)}$ in the deepest layer of a DeepCDM are modeled using a categorical distribution:

(6) $$ \begin{align} \mathbb P(\mathbf A^{(D)} = \boldsymbol \alpha_{\ell}) &= \pi^{(D)}_{\boldsymbol \alpha_{\ell}}, \quad \forall \boldsymbol \alpha_{\ell} \in \{0,1\}^{K_D}. \end{align} $$

Here, $K_D> 1$ . The proportion parameters $\pi ^{(D)}_{\boldsymbol \alpha _{\ell }}$ are subject to the constraint $\sum _{\boldsymbol \alpha _{\ell } \in \{0,1\}^{K_D}} \pi ^{(D)}_{\boldsymbol \alpha _{\ell }} = 1$ . With this, we complete the specification of a general DeepCDM.

We next discuss the distinction between the proposed DeepCDM and the attribute hierarchy method (AHM; Gierl et al., Reference Gierl, Leighton, Hunka and Cambridge2007; Templin & Bradshaw, Reference Templin and Bradshaw2014). Both approaches aim to capture dependencies among latent attributes. However, they differ fundamentally in how such dependencies are represented and inferred. In the AHM framework, hierarchies are defined as deterministic prerequisite relations among attributes within a single latent layer—for instance, mastering attribute B is a prerequisite for mastering attribute A. These relations constrain admissible attribute profiles to those consistent with the specified hierarchy, embedding learning sequences informed by substantive or curricular theory. In contrast, DeepCDM introduces a probabilistic hierarchy across multiple latent layers. Rather than specifying logical prerequisites within one layer, it models how attributes at one layer are statistically explained by latent traits at a deeper and higher-order layer. Higher-order traits capture shared variance among lower-level attributes, enabling a data-driven hierarchical representation learned directly from data. Overall, these distinctions indicate that AHM and DeepCDM reflect different modeling perspectives and are therefore suited to different types of applications.

2.2 Specific examples of DeepCDMs

This section presents concrete examples of DeepCDMs that fall under the general framework outlined in Section 2.1. For notational convenience, we also denote the observed response layer $\mathbf R$ as $\mathbf A^{(0)}$ , enabling a unified expression for the layerwise conditional distributions: $\mathbb {P}(\mathbf A^{(d-1)} \mid \mathbf A^{(d)}, \mathbf Q^{(d)})$ for $d = 1, \dots , D$ . We then define specific DeepCDM variants according to the diagnostic model adopted for each layerwise conditional.

Example 1 (Main-effect DeepCDMs).

We use the term “Main-effect DeepCDMs” to refer broadly to DeepCDMs in which each layerwise conditional distribution follows a main-effect diagnostic model. In this setup, the probability that $A^{(d-1)}_j = 1$ is governed by the main effects of its parent attributes, modeled via a link function $f(\cdot )$ :

(7) $$ \begin{align} \mathbb{P}(A^{(d-1)}_j = 1 \mid \mathbf A^{(d)} = \boldsymbol \alpha, \mathbf Q^{(d)}, \boldsymbol \beta^{(d)}) = f\Big( \beta^{(d)}_{j,0} + {\sum}_{k=1}^{K_d} \beta^{(d)}_{j,k} \left\{q^{(d)}_{j,k} \alpha_k \right\} \Big). \end{align} $$

Here, $\beta _{j,k}^{(d)}$ is nonzero only when $q_{j,k}^{(d)} = 1$ . When f is the identity function, Equation (7) reduces to the additive CDM (ACDM; de la Torre, Reference de la Torre2011). If f is the inverse logit function, Equation (7) gives a logistic linear model (LLM; Maris, Reference Maris1999).

Example 2 (All-effect DeepCDMs).

We refer to DeepCDMs in which the layerwise conditionals follow an all-effect diagnostic model as “All-effect DeepCDMs.” In an all-effect diagnostic model, the probability of $A^{(d-1)}_j = 1$ depends on both the main effects and all possible interaction effects of the parent attributes:

(8) $$ \begin{align} & \mathbb{P}(A^{(d-1)}_j = 1 \mid \mathbf A^{(d)} = \boldsymbol \alpha, \mathbf Q^{(d)}, \boldsymbol \beta^{(d)}) = f\Big( \beta^{(d)}_{j,0} + {\sum}_{k=1}^{K_d} \beta^{(d)}_{j,k} \left\{q^{(d)}_{j,k} \alpha_k \right\} \\ \notag &\qquad\qquad\qquad + {\sum}_{1 \leq k_1 < k_2 \leq K_d} \beta^{(d)}_{j,k_1 k_2} \left\{q^{(d)}_{j,k_1} \alpha_{k_1}\right\} \left\{q^{(d)}_{j,k_2} \alpha_{k_2}\right\} + \cdots + \beta^{(d)}_{j,12 \dots K_d} {\prod}_{k=1}^{K_d} \left\{q^{(d)}_{j,k} \alpha_k\right\} \Big). \end{align} $$

Similar to the main-effect model, not all $\beta $ -coefficients above are needed. If $\boldsymbol q^{(d)}_j$ , the j-th row of $\mathbf {Q}^{(d)}$ , contains $K_j$ entries of “1,” then $2^{K_j}$ parameters are required in (8). With the identity link function, (8) defines the generalized DINA model (GDINA; de la Torre, Reference de la Torre2011), while the inverse logit function yields the log-linear CDM (LCDM; Henson et al., Reference Henson, Templin and Willse2009).

Example 3 (DeepDINA).

The DINA model can be regarded as a special case of the all-effect CDM, where only the highest-order interaction term among the required attributes is retained, and all lower-order effects are constrained to zero:

(9) $$ \begin{align} \mathbb{P}^{\mathrm{DINA}}(A^{(d-1)}_j = 1 \mid \mathbf A^{(d)} = \boldsymbol \alpha, \mathbf Q^{(d)}, \boldsymbol \beta^{(d)}) = f\Big(\beta_{j,0}^{(d)} + \beta_{j,\mathcal{K}^{(d)}_j} \prod_{k \in {\mathcal{K}^{(d)}_j}} q^{(d)}_{j,k}\alpha_k \Big), \end{align} $$

where ${\mathcal {K}^{(d)}_j} = \left \{k \in [K];\, q_{jk}^{(d)} = 1\right \}$ denotes the set of attributes measured by item j. The model assumes that students are capable of an item only if they master all required attributes for that item. So, $\beta _{j,\mathcal {K}^{(d)}_j}$ is the only non-zero coefficient for item j in layer d.

One can also specify a DeepDINO model, a specific type of DeepCDM where the DINO model is used to model each latent layer (Gu, Reference Gu2024). Due to the duality between DINA and DINO, the identifiability and algorithm applicable to DeepDINA are also applicable to DeepDINO. Therefore, we do not introduce it here and refer readers to Gu (Reference Gu2024) for details.

As demonstrated earlier, only particular coefficients, determined by the $\mathbf Q$ -matrices and the specified measurement models, in the generating DeepCDM should be non-zero. However, since all $\mathbf Q$ -matrices, $\mathbf Q^{(d)}, d = 1, \dots , D$ , are unknown, the sparsity pattern of the coefficient vectors is also unknown. Therefore, we assume all coefficients in the model as unknown and estimate them by maximizing a regularized log-likelihood. The $\mathbf Q$ -matrices can then be inferred by identifying the non-zero coefficients in $\boldsymbol {\beta }^{(d)}$ , $d=1,\ldots ,D$ . We defer the details of the mechanism for identifying the entries $q_{jk}^{(d)}$ to Section 3.2.

2.3 DeepCDMs as deep generative models

As previously mentioned, the DeepCDM framework can be viewed as an adaptation of DGMs for psychometrics and educational measurement, where additional structural constraints are introduced to enable diagnostic feedback. Exploratory DeepCDMs share architectural similarities with several existing DGMs. For example, when the activation function $ f $ is defined as the inverse logit, DeepCDMs resemble DBNs (Hinton et al., Reference Hinton, Osindero and Teh2006) with binary-valued hidden units. However, a key structural difference lies at the top of the network: DBNs assume an undirected graph between the top two layers—forming a restricted Boltzmann machine (RBM)—while DeepCDMs adopt a fully directed, top-down architecture across all layers. This design enables DBNs to use a heuristic greedy layer-wise pretraining procedure based on contrastive divergence, a technique specific to training undirected models, such as RBMs (Hinton & Salakhutdinov, Reference Hinton and Salakhutdinov2006; Hinton et al., Reference Hinton, Osindero and Teh2006). Consequently, such training strategies are not directly applicable to DeepCDMs due to its directed nature, which is more interpretable for modeling hierarchical skill generation.

DeepCDMs also share a top-down generative structure with DEFs (Ranganath et al., Reference Ranganath, Tang, Charlin and Blei2015), an unsupervised framework using exponential family distributions to model each layer’s conditional distribution. DEFs aim to capture compositional semantics through hierarchical latent representations. However, DEFs rely on black-box variational inference with neural network-based posterior approximations, which prevent recovery of interpretable parameters, such as $\mathbf {Q}$ -matrices, and thus cannot provide individualized diagnostic feedback, a central aspect of cognitive diagnosis.

Another related framework is the deep discrete encoders (DDEs; Lee & Gu, Reference Lee and Gu2025), a DGM designed for rich data types with discrete latent layers. While DDEs and DeepCDMs share architectural and identifiability properties, their goals differ. DDEs aim to address machine learning concerns like overparameterization and lack of interpretability, constructing general-purpose identifiable DGMs. In contrast, DeepCDMs are specifically designed for psychometrics, with each adjacent pair of latent layers constituting a CDM. This structure allows for diagnostic-specific measurement assumptions, addressing varied diagnostic goals and enhancing usability in real-world assessment settings.

Taken together, these distinctions highlight the unique features of exploratory DeepCDMs over existing DGMs. While most DGMs prioritize data generation or predictive performance and typically lack identifiability and sparsity constraints, DeepCDMs refocus the modeling effort on offering reliable individualized diagnostic feedback and discovering hierarchical latent skill structures. Furthermore, by enforcing sparsity in the coefficient matrices to reflect item–attribute relationships, DeepCDMs provide valuable insights into test design—an essential feature for practical use in educational and psychological assessment, often overlooked in classical DGM frameworks.

2.4 Identifiability

As noted earlier, a key strength of DeepCDMs lies in their formal identifiability guarantees, which apply to both confirmatory and exploratory settings (Gu, Reference Gu2024). These results are detailed in the Supplementary Material. In brief, the identification conditions impose explicit structural constraints on the between-layer $\mathbf Q$ -matrices, offering practical guidance for model design and implementation. Although the specific conditions vary across diagnostic models, they consistently require an increasingly shrinking latent structure for deeper layers. That is, the number of latent variables typically decreases with depth, often subject to constraints, such as $K_d> c \cdot K_{d+1}$ for some constant $c> 1$ depending on the model. This hierarchy reflects the principle of statistical parsimony in DeepCDMs. For instance, in a two-layer DeepDINA model with $K_1=7$ and $K_2=2$ , the number of nonzero parameters is $2K_1 + 2^{K_2} - 1 = 17$ , compared to $2^{K_1} - 1 = 127$ in a saturated attribute model without higher-order latent structure. Such substantial reductions in complexity make DeepCDMs especially attractive for applications with fine-grained latent attributes and limited sample sizes. In exploratory settings, while all parameters must be estimated, the identifiability conditions naturally promote sparsity in the true generating model, facilitating parameter recovery and interpretation of the latent attributes.

A central insight underlying these proofs is that the identifiability of DeepCDMs can be established in a layer-by-layer fashion, proceeding from the bottom (shallow) layer to the top (deepest) layer. This approach is justified by the directed nature of the graphical model and the discreteness of latent variables. Two core ideas facilitate this stepwise identifiability. First, in a multi-layer directed graphical model with only top-down connections between adjacent layers, marginalizing out deeper latent layers yields a restricted latent class model (RLCM) (Gu & Xu, Reference Gu and Xu2020; Xu, Reference Xu2017). Once the distribution of the shallowest latent layer is identified through this RLCM, it can be treated theoretically as if observed for identifying the next deeper layer. Second, identifiability in RLCMs holds for any marginal distribution of latent attributes, provided the $\mathbf Q$ -matrix meets specific conditions. This property allows identifiability to propagate upward through layers, even when deeper latent variables introduce complex dependencies.

3.1 The EM algorithm

Let $\mathbf A_{1:N} = (\mathbf A^{(1)}_{1:N}, \ldots , \mathbf A^{(D)}_{1:N})$ denote the set of latent variables, i.e., the attribute profiles of the N students across D latent layers. The complete data log-likelihood is:

(12) $$ \begin{align} &l_c^{DeepCDM} (\boldsymbol{\beta}, \boldsymbol{\pi}^{(D)},\mathbf Q|\mathbf R_{1:N}, \mathbf A_{1:N})\nonumber\\ =& \log\left( \mathbb{P}(\mathbf R_{1:N}\mid \mathbf A^{(1)}_{1:N},\mathbf Q^{(1)},\boldsymbol{\beta}^{(1)}) \cdot \prod_{d=2}^{D} \mathbb{P}(\mathbf A^{(d-1)}_{1:N}\mid \mathbf A^{(d)}_{1:N},\mathbf Q^{(d)},\boldsymbol{\beta}^{(d)}) \cdot \mathbb{P}(\mathbf A^{(D)}_{1:N};\boldsymbol{\pi}^{(D)})\right). \end{align} $$

Let $\boldsymbol {\beta }^{(t-1)}=(\boldsymbol {\beta }^{(1,t-1)}, \ldots , \boldsymbol {\beta }^{(D,t-1)})$ , $\boldsymbol {\pi }^{(D,t-1)}$ , and $\mathbf Q^{(t-1)} = (\mathbf Q^{(1,t-1)}, \ldots , \mathbf Q^{(D,t-1)})$ denote the estimates of $\boldsymbol {\beta }$ , $\boldsymbol {\pi }^{(D)}$ , and $\mathbf Q$ obtained at iteration $t-1$ . In each iteration t of the EM algorithm, the following two steps are performed:

E-Step: Compute

(13) $$ \begin{align} \widetilde Q^{(t)}=\mathbb{E}_{(\mathbf A^{(1)}, \ldots, \mathbf A^{(D)})}[l_c^{DeepCDM} (\boldsymbol{\beta},\boldsymbol{\pi}^{(D)},\mathbf Q|\mathbf R_{1:N}, \mathbf A_{1:N})|\mathbf R_{1:N};\boldsymbol{\beta}^{(t-1)},\boldsymbol{\pi}^{(D,t-1)},\mathbf Q^{(t-1)}], \end{align} $$

where the conditional expectation is with respect to $\mathbb {P}(\mathbf A^{(1)}, \ldots , \mathbf A^{(D)}|\mathbf R_{1:N};\boldsymbol {\beta }^{(t-1)},\boldsymbol {\pi }^{(D,t-1)},\mathbf Q^{(t-1)})$ .

M-Step: Update

(14) $$ \begin{align} \left(\boldsymbol{\beta}^{(t)},\boldsymbol{\pi}^{(D,t)},\mathbf Q^{(t)}\right) &= \underset{\boldsymbol{\beta},\mathbf Q,{\boldsymbol{\pi}^{(D)}}} {\text{arg max}} \,\widetilde Q^{(t)}-N\cdot P_{\boldsymbol{s}}(\boldsymbol{\beta}), \end{align} $$

where $P_{\boldsymbol {s}}(\boldsymbol {\beta })$ is defined in Equation (11).

For each $d=1,2,\ldots ,D$ , define $\widetilde {\mathbf A}^{1:d}=(\mathbf A^{(1)},\ldots ,\mathbf A^{(d)})$ , which denote the latent variables shallower than the $(d+1)$ th latent layer. Define $\mathbf A^{(0)}=\mathbf {R}_{1:N}$ and $\mathbf A^{(D+1)}=\emptyset $ . According to the conditional independence, the expectation computation in the E-step can be re-expressed as $\widetilde Q^{(t)}= \sum _{d=1}^{D+1} \widetilde {Q}^{(d,t)},$ with

(15) $$ \begin{align} \widetilde{Q}^{(d,t)}=&\mathbb E_{\widetilde{\mathbf A}^{1:d}} [\log{P(\mathbf A^{(d-1)}_{1:N}|\mathbf A^{(d)}_{1:N})}|\mathbf R_{1:N},\boldsymbol{\beta}^{(t-1)},\boldsymbol{\pi}_D^{(t-1)},\mathbf Q^{(t-1)}]\nonumber\\ &=\sum_{i=1}^N \mathbb E_{\widetilde{\mathbf A}^{1:d}} [\log{P(\mathbf A^{(d-1)}|\mathbf A^{(d)})}|\mathbf R_{i},\boldsymbol{\beta}^{(t-1)},\boldsymbol{\pi}_D^{(t-1)},\mathbf Q^{(t-1)}] \end{align} $$

That is, $\widetilde Q^{(t)}$ is decomposed as a summation over layers d and individuals i, where each term is the conditional expectation of $\log P(\mathbf A^{(d-1)} \mid \mathbf A^{(d)})$ with respect to the partial posterior distribution $P(\mathbf A^{(1)}, \ldots , \mathbf A^{(d)} \mid \mathbf R_i, \boldsymbol {\beta }^{(t-1)}, \boldsymbol {\pi }_D^{(t-1)}, \mathbf Q^{(t-1)})$ , which we denote by $\widetilde P^{1:d}_{i}$ for brevity.

Accordingly, the optimization in the M-step can be broken into the following parts:

(16) $$ \begin{align} (\boldsymbol{\beta}^{(d,t)},\mathbf Q^{(d,t)})&= \underset{\boldsymbol{\beta}^{(d)},\mathbf Q^{(d)}}{\text{arg max}} \,\widetilde{Q}^{(d,t)}-N\cdot P_{\boldsymbol{s}}(\boldsymbol{\beta}^{(d)}),\quad d=1,\ldots,D. \end{align} $$

(17) $$ \begin{align} \boldsymbol{\pi}^{(D,t)}&= {\text{arg max}} \sum_{\boldsymbol \alpha^{(D)}}\sum_{i=1}^N \log(\mathbb{P}(\mathbf A^{(D)}=\boldsymbol \alpha^{(D)})) \times\nonumber\\ \sum_{(\boldsymbol \alpha^{(1)},\ldots,\boldsymbol \alpha^{(D-1)})} & P\left(\mathbf A_i^{(1)}=\boldsymbol \alpha^{(1)}, \mathbf A_i^{(2)}=\boldsymbol \alpha^{(2)},\ldots, \mathbf A_i^{(D)}=\boldsymbol \alpha^{(D)}|\mathbf R_{i};\boldsymbol{\beta}^{(t-1)},\boldsymbol{\pi}_D^{(t-1)},\mathbf Q^{(t-1)}\right). \end{align} $$

This decomposition enables the parameters at each layer to be updated via their corresponding optimization problems in (16) and (17), thereby improving the tractability of the M-step. Next, we further look into the $\widetilde {Q}^{(d,t)}$ functions. Recall that $\widetilde P^{1:d}_{i}$ is defined as

(18) $$ \begin{align} \widetilde P^{1:d}_{i} := P(\widetilde{\mathbf A}^{1:d} \mid \mathbf R_{i}, \boldsymbol{\beta}^{(t-1)}, \boldsymbol{\pi}_D^{(t-1)}, \mathbf Q^{(t-1)}) = \frac{P(\mathbf R_{i} \mid \mathbf A^{(1)}) P(\widetilde{\mathbf A}^{1:d})}{\sum_{\widetilde{\mathbf A}^{1:d}} P(\mathbf R_{i} \mid \mathbf A^{(1)}) P(\widetilde{\mathbf A}^{1:d})}, \end{align} $$

with

(19) $$ \begin{align} P(\widetilde{\mathbf A}^{1:d}) = \sum_{\boldsymbol \alpha^{(d+1)},\ldots,\boldsymbol \alpha^{(D)}} &P\left(\mathbf A^{(1)}, \ldots, \mathbf A^{(d)} \mid \mathbf A^{(d+1)} = \boldsymbol \alpha^{(d+1)}, \boldsymbol{\beta}^{(t-1)} \right) \times \nonumber \\ &P\left(\mathbf A^{(d+1)} = \boldsymbol \alpha^{(d+1)}, \ldots, \mathbf A^{(D)} = \boldsymbol \alpha^{(D)}; \boldsymbol{\beta}^{(t-1)} \right). \end{align} $$

As shown, given the parameters from the previous iteration $(t-1)$ , the distribution $P(\widetilde {\mathbf A}^{1:d})$ is computed by marginalizing out the deeper latent variables $\mathbf A^{(d+1)}, \dots , \mathbf A^{(D)}$ from the joint distribution over all latent variables. This marginal, together with the observed response $\mathbf R_i$ , allows us to evaluate the partial posterior distribution $\widetilde {P}^{1:d}_{i}$ via Equation (18). With $\widetilde {P}^{1:d}_{i}$ providing the weights for each possible configuration of $\widetilde {\mathbf A}^{1:d}$ , the conditional expectation in $\widetilde {Q}^{(d,t)}$ can be computed as a weighted sum over $\log P(\mathbf A^{(d-1)} \mid \mathbf A^{(d)})$ . Specifically, Equation (15) can be written out as

(20) $$ \begin{align} \widetilde{Q}^{(1,t)} = \sum_{i=1}^N \sum_{\mathbf{A}^{(1)}} \log P(\mathbf R_i|\mathbf{A}^{(1)}) \widetilde{P}^{1:1}_{i}; \end{align} $$

and

(21)

for $d=2,\ldots ,D$ . It turns out that, for each d, Equation (16) defines a regularized optimization problem whose objective includes a weighted log-likelihood component over $(\mathbf {A}^{(d-1)}, \mathbf {A}^{(d)})$ pairs. In this formulation, $\mathbf {A}^{(d-1)}$ serves as the outcome, $\mathbf {A}^{(d)}$ as the predictor, and the data point weights are given by $\sum _{i=1}^N \sum _{\widetilde {\mathbf A}^{1:d}\backslash \left \{\mathbf {A}^{(d-1)},\mathbf {A}^{(d)}\right \}} \widetilde {P}^{1:d}_{i}$ .

We focus on the case where the link function $ f(\cdot ) $ is the inverse logit, as it is the most commonly used choice for CDMs with binary responses. In this setting, the estimation problem corresponds to a generalized linear optimization problem with a logit link. For other choices of $ f(\cdot ) $ , the problem may fall into the broader categories of linear or generalized linear optimization, depending on the specific functional form. The solution of Equation (17) is that for $\forall \boldsymbol \alpha ^{(D)}\in \{0,1\}^{K_D}$ :

(22) $$ \begin{align} \boldsymbol{\pi}_{\boldsymbol \alpha^{(D)}}^{(D,t)}=\sum_{i=1}^N \sum_{\mathbf A^{(1)},\ldots, \mathbf A^{(D-1)}}\frac{P(\mathbf R_{i}|\mathbf A^{(1)})P(\mathbf A^{(1)},\ldots,\mathbf A^{(D-1)},\mathbf A^{(D)}=\boldsymbol \alpha^{(D)})}{N\cdot\sum_{\mathbf A^{(1)},\ldots, \mathbf A^{(D)}}P(\mathbf R_{i}|\mathbf A^{(1)})P(\mathbf A^{(1)},\ldots,\mathbf A^{(D-1)},\mathbf A^{(D)})}. \end{align} $$

These derivations demonstrate that, due to conditional independence, the EM algorithm for exploratory DeepCDMs is both succinct and transparent. However, its practical feasibility is challenged by several issues—particularly sensitivity to initialization and the accumulation of estimation bias across layers and iterations. To address these challenges, we next propose a layer-wise EM algorithm below.

3.2 A novel layer-wise EM algorithm

To elucidate the underlying rationale of the proposed algorithm, we first provide a detailed mathematical derivation of the layer-wise EM procedure. This step-by-step formulation highlights how the algorithm naturally arises from the hierarchical structure of DeepCDMs.

Suppose we have a set of parameters $ \boldsymbol {\beta }, \boldsymbol {\pi }^{(D)}, \mathbf Q $ that maximize the regularized marginal log-likelihood in Equation (10). Based on the generative formulation of DeepCDMs, we can marginalize out the deeper latent variables $ \mathbf A^{(2)}, \ldots , \mathbf A^{(D)} $ to derive the implied distribution of the bottom-layer attributes:

$$\begin{align*}\boldsymbol{\pi}^{(1)} = \mathbb{P}(\mathbf A^{(1)}; \boldsymbol{\pi}^{(1)}) = \sum_{\boldsymbol \alpha^{(2)}} \cdots \sum_{\boldsymbol \alpha^{(D)}} \Big[ \prod_{d=2}^{D} \mathbb{P}(\mathbf A^{(d-1)} \mid \mathbf A^{(d)}, \mathbf Q^{(d)}, \boldsymbol{\beta}^{(d)}) \cdot \mathbb{P}(\mathbf A^{(D)}; \boldsymbol{\pi}^{(D)}) \Big]. \end{align*}$$

This recursive marginalization induces a set of shallow-layer parameters $ (\boldsymbol {\beta }^{(1)}, \boldsymbol {\pi }^{(1)}, \mathbf Q^{(1)}) $ , which, when substituted into the original model, must also maximize a re-expressed form of the target function:

$$ \begin{align*} \ell(\boldsymbol{\beta}^{(1)}, \boldsymbol{\pi}^{(1)}, \mathbf Q \mid \mathbf R_{1:N}) = \sum_{i=1}^N \log \Bigg\{ \sum_{\boldsymbol \alpha^{(1)}} \Big[ \mathbb{P}(\mathbf R_i \mid \mathbf A^{(1)}, \mathbf Q^{(1)}, \boldsymbol{\beta}^{(1)}) \cdot \mathbb{P}(\mathbf A^{(1)}; \boldsymbol{\pi}^{(1)}) \Big] \Bigg\} - N \cdot P_{\boldsymbol{s}}(\boldsymbol{\beta}^{(1)}). \end{align*} $$

This observation forms the conceptual basis of our proposed layer-wise EM algorithm. Rather than estimating all parameters jointly over the entire deep latent architecture, we decompose the problem into a sequence of simpler subproblems, each involving a one-layer structure, and solve them in a bottom-up manner using EM. Although the resulting algorithm appears intuitive, it is grounded in a rigorous use of the model’s generative structure. Specifically, each layer-wise step leverages the most reliable information from its immediate lower layer—either in the form of estimated distributions or pseudo-observations—making the estimation process both computationally efficient and statistically reliable.

Focusing on the first layer ( $ d = 1 $ ), this decomposition implies that the estimates $ \boldsymbol {\beta }^{(1)} $ and $ \mathbf Q^{(1)} $ obtained by maximizing Equation (10) are identical to those obtained under a standard one-layer CDM. Once these first-layer parameters are estimated, the task reduces to estimating the parameters for the remaining $ D-1 $ latent layers. Unlike the first layer, however, there are no observed realizations of the latent variable $ \mathbf A^{(1)} $ . Let $ \mathbf A^{(1)}_i $ denote the $ i $ -th row of $ \mathbf A^{(1)}_{1:N} $ . A straightforward way to impute $ \mathbf A^{(1)}_i $ is via the maximum a posteriori (MAP) estimate: $ \widehat {\mathbf A}^{(1)}_i = \arg \max _{\boldsymbol \alpha ^{(1)} \in \{0,1\}^{K_1}} \mathbb {P}(\mathbf A^{(1)}_i = \boldsymbol \alpha ^{(1)} \mid \mathbf R_i, \boldsymbol {\beta }^{(1)}, \mathbf Q^{(1)}), $ which depends on both the likelihood $ \mathbb {P}(\mathbf R_i \mid \mathbf A^{(1)}_i, \boldsymbol {\beta }^{(1)}, \mathbf Q^{(1)}) $ and the prior $ \mathbb {P}(\mathbf A^{(1)}_i) $ . Alternatively, one can sample from the estimated marginal distribution $ \mathbb {P}(\mathbf A^{(1)}) $ to generate pseudo-observations for the next layer. Compared to MAP, this sampling-based approach introduces less bias and leads to more reliable parameter estimation in deeper layers, particularly during initialization. With these pseudo-observations in hand, the remaining $ D-1 $ layers form a structurally similar DeepCDM, and the same one-layer EM algorithm can be recursively applied to estimate $ \boldsymbol {\beta }^{(2)} $ and $ \mathbf Q^{(2)} $ , and so on, until the top layer is reached.

The above derivation implies that the estimator obtained from the layer-wise procedure is statistically equivalent to the estimator obtained by jointly maximizing the full DeepCDM likelihood, and thus achieves the same asymptotic efficiency. This procedure offers practical computational advantages, as discussed in Section 3.4. Although the estimation at layer d uses only the latent variables from the immediately deeper layer $(d{+}1)$ , this does not discard the dependencies encoded in all deeper layers. Indeed, that information is summarized in the aggregated prior $\boldsymbol {\pi }^{(d+1)}$ , which captures the deeper dependency structure and allows the procedure at layer d to focus on recovering the relationship between $\mathbf A^{(d)}$ and $\mathbf A^{(d+1)}$ . By iterating this process upward through the hierarchy, the estimated parameters $(\boldsymbol {\beta }^{(d)}, \mathbf {Q}^{(d)}, \boldsymbol {\pi }^{(d)})$ across all layers collectively recover the full cross-layer dependence structure.

Algorithm 1 summarizes the proposed layer-wise EM algorithm. Proceeding from the bottom up, the algorithm estimates model parameters layer by layer. For each layer $ d> 1 $ , Step 1 imputes the missing data $ \mathbf A^{(d-1)} $ by drawing samples from the estimated marginal distribution $ P(\mathbf A^{(d-1)}) $ , which is computed in Step 3 of the previous layer using Equation (23). In contrast, for the first latent layer ( $ d = 1 $ ), Step 1 is not required because the observed response data $ \mathbf R_{1:N} $ (i.e., $ \mathbf A^{(0)} $ ) serve directly as the input. Using the initial values obtained in Step 2, Step 3 then applies a one-layer EM algorithm with $ K_d $ attributes to estimate the parameters $ \boldsymbol {\beta }^{(d)} $ , $ \mathbf Q^{(d)} $ , and $ \boldsymbol {\pi }^{(d)} $ . The initialization procedure is introduced separately in Section 3.3. In Step 3, each M1-step is solved using the coordinate descent algorithm of Friedman et al. (Reference Friedman, Hastie and Tibshirani2010), known for its flexibility and effectiveness in handling regularized optimization problems. The algorithm continues this layer-wise procedure until reaching the deepest layer $ D $ .

In M2-step, the mechanism for identifying the entries $q_{jk}^{(d)}$ in $\mathbf Q^{(d)}$ varies across different measurement models, according to the measurement model utilized in layer d. For the main-effect-model, $\mathbf Q^{(d)}$ can be recovered using the rule $ q_{jk}^{(d)} = \boldsymbol {1}(\beta _{j,k}^{(d)} \neq 0) $ , where $ \boldsymbol {1} $ is the indicator function. For the all-effect model, theoretically, each row of $\mathbf Q^{(d)}$ should be identified by the highest-order non-zero coefficient. Specifically, if $ \exists S \subseteq [K] $ such that $ \beta _{j,S}^{(d)} \neq 0 $ and $ \beta _{j,S'}^{(d)} = 0 $ for all $ S' \subseteq [K], S \subset S' $ , then $ q_{jk}^{(d)} = 1 $ for $ k \in S $ ; otherwise, $ q_{jk}^{(d)} = 0 $ . However, this strict rule may not always be applicable because some estimated ${\beta }$ -coefficients may be close to zero but not exactly zero due to residual regularization noise. In practice, a more effective approach is either to choose the largest non-zero interaction coefficient or to truncate the coefficients before identifying $\mathbf Q^{(d)}$ . For the latter approach, we recommend practitioners set the truncation thresholds based on the general magnitude of their estimated coefficients. For the DINA model, since there should be only one non-zero coefficient for each item $ j $ , the largest non-zero interaction coefficient can be selected, and the corresponding $ q_{jk}^{(d)} $ can be identified as equal to one. We note that seemingly poor $\mathbf Q$ -matrix recovery may appear if such residual regularization noise is not properly addressed, as it can obscure the algorithm’s underlying recovery performance.

3.3 Initialization algorithm

As shown in Algorithm 1, the initialization of our DeepCDM method is performed sequentially in D steps, where parameters from the d-th latent layer are initialized after estimating the $(d-1)$ -th layer. This approach leverages the information from the estimated distribution $P(\mathbf {A}^{(d-1)})$ , obtained during the fitting of the $(d-1)$ -th layer, to provide better initial values for the d-th layer. Furthermore, by imputing realizations of $\mathbf {A}^{(d-1)}$ sampled from the estimated $P(\mathbf {A}^{(d-1)})$ , the initializations of all layers reduce to the problem of initializing a one-layer CDM. Specifically, the response data $\mathbf {R}_{1:N}$ are used for the first layer ( $d=1$ ), while realizations of $\mathbf {A}^{(d)}$ (i.e., the generated pseudo-samples) are used for subsequent layers ( $d>1$ ). For exploratory DeepCDMs, good initial values should not only be close to the true values but also exhibit a sparse structure similar to the true ones. To achieve this, we apply a USVT-based method (Chatterjee, Reference Chatterjee2015; Zhang et al., Reference Zhang, Chen and Li2020) to estimate the design matrix, followed by a Varimax rotation to promote sparsity. USVT captures the dominant low-rank structure, while Varimax produces a sparse loading pattern that informs the initial $\mathbf {Q}$ -matrix. This combination provides informative starting values tailored to the exploratory framework of DeepCDMs. The initialization procedure for each layer d is detailed in Algorithm 2.

Algorithm 2 applies SVD twice. The first application, combined with the inverse transformation (Steps 2–5), is used to denoise and linearize the data. The second application of SVD (Steps 6 and 7) is performed on the linearized data. Since SVD solutions are rotation-invariant while the true coefficient matrix $\boldsymbol {\beta }_1^{(d)}$ is sparse, we address this invariance by seeking a sparse representation of the denoised data that is consistent with the model’s identifiability conditions and expect it to share the same sparsity pattern as $\boldsymbol {\beta }_1^{(d)}$ . We apply the Varimax rotation for this purpose in Step 8, as it induces sparse loadings and possesses well-established statistical inference properties (Rohe & Zeng, Reference Rohe and Zeng2023). To further refine sparsity, a thresholding step is applied to eliminate near-zero loadings resulting from sampling variability (Step 9). Rather than using the conventional threshold $1/\sqrt {N}$ , we adopt a more relaxed threshold $1/(2\sqrt {K_{d-1}})$ , which is more adaptive to the multilayer structure of DeepCDMs—since pseudo-data in deeper layers tend to be noisier, a more relaxed threshold helps mitigate such effects. If the rotated Varimax loadings were used directly to estimate $\mathbf A^{(d)}$ and $\boldsymbol {\beta }_1^{(d)}$ , the estimates could differ from the true ones in scale, as Varimax does not constrain the scaling of rows or columns. For $\mathbf A^{(d)}$ , although this scaling issue remains, the binary structure can still be identified by noting that $A(i,k)=0$ if and only if $A_0(i,k)<0$ in Step 10. We then exploit the discreteness of the latent variables in $\mathbf A^{(d)}$ to rescale $\boldsymbol {\beta }_1^{(d)}$ in Step 11, yielding correctly scaled estimates. This initialization procedure is non-iterative, making it computationally efficient. Moreover, it avoids convergence issues and possesses favorable statistical consistency properties (Zhang et al., Reference Zhang, Chen and Li2020).

Additionally, as pointed out by one reviewer, one can also directly apply the Varimax rotation to the matrix $ \frac {1}{\sqrt {N}} (\widehat {\tau }_1 \widehat {v}_1, \ldots , \widehat {\tau }_{K_{d-1}} \widehat {v}_{K_{d-1}}). $ A simulation study comparing the performance of these two approaches is reported in the Supplementary Material.

3.4 Advantages of layer-wise EM compared to EM

As discussed in Hinton et al. (Reference Hinton, Osindero and Teh2006), an effective way to learn complex models is by sequentially combining simpler models. The layer-wise EM algorithm applies this principle by decomposing the optimization into manageable subproblems, each targeting one layer at a time. This strategy helps overcome key challenges faced by the classical EM algorithm when applied to DeepCDMs.

One major challenge in applying the classical EM algorithm to DeepCDMs is the difficulty of initialization. The presence of multiple nonlinear latent layers creates a highly nonconvex optimization landscape, often with an exponential number of local optima. As a result, EM is sensitive to starting values—poor initialization can easily lead to convergence at suboptimal local maxima of the penalized log-likelihood function. In the classical EM algorithm, initial values for all parameters across all layers must be specified simultaneously. As the model depth increases, this becomes increasingly challenging, even for moderately deep architectures, due to compounded uncertainty and parameter interactions across layers. In contrast, our layer-wise EM algorithm addresses initialization sequentially by solving one-layer CDMs one at a time. At each stage, it initializes the parameters of the current layer using either observed responses or sampled latent attributes from the estimated marginal distribution informed by the shallower layers. Since these samples are based on estimated proportion parameters which are proved to be identifiable, the initialization for deeper layers is more stable and reliable.

Another major issue of the classical EM algorithm is the accumulation of estimation bias as the algorithm progresses through multiple layers. As the number of latent layers $ D $ increases, the computation of quantities like $ \sum _{i=1}^N \widetilde {P}^{1:d}_{i} $ for $ d = 1, \ldots , D $ becomes more error-prone. These quantities play a crucial role in the M-step, and inaccuracies in their estimation directly affect parameter updates. Furthermore, the iterative nature of the classical EM algorithm introduces a cyclic dependency across all layers, where errors at one layer can propagate forward and backward, reinforcing one another over iterations. This compounding effect often leads the algorithm to converge to suboptimal local maxima, even when reasonably good initial values are provided. In contrast, our proposed layer-wise EM algorithm mitigates this issue by breaking the dependency cycle. Parameters are estimated sequentially from the bottom layer up, so each layer $ d $ is only influenced by the estimates from shallower layers $ d' < d $ . Although some bias may still accumulate, it originates solely from estimation errors in previous layers—not from compounded initialization errors across all layers. This directional, non-cyclic structure significantly reduces the accumulation of error and enhances the robustness of the estimation process.

3.5 Connections to broader principles and algorithms

3.5.1 Connections between the layer-wise EM and identifiability

Interestingly, the derivation of the layer-wise EM algorithm aligns with and supports the technical insights in the identifiability proofs of DeepCDMs. The identifiability results in the Supplementary Material and in Gu (Reference Gu2024) demonstrate that the identifiability of a DeepCDM can be examined and established in a layer-by-layer manner, proceeding from the bottom up. This follows from the probabilistic formulation of the directed graphical model and the discrete nature of the latent layers. For each layer $ d \in \left \{0,1,\ldots , D-1\right \} $ , as long as $ \mathbf Q^{(d+1)} $ satisfies the corresponding identifiability conditions for the one-layer CDM implied by

(24) $$ \begin{align} \mathbb{P}(\mathbf A^{(d)}) = \sum_{\boldsymbol \alpha^{(d+1)} \in \{0,1\}^{K_{d+1}}} \mathbb{P}(\mathbf A^{(d)} \mid \mathbf A^{(d+1)} = \boldsymbol \alpha^{(d+1)}, \mathbf Q^{(d+1)}, \boldsymbol{\theta}^{(d+1)}) \cdot \mathbb{P}(\mathbf A^{(d+1)} = \boldsymbol \alpha^{(d+1)}), \end{align} $$

the parameter set $ (\boldsymbol {\beta }^{(d+1)}, \mathbf Q^{(d+1)}, \boldsymbol {\pi }^{(d+1)}) $ is identifiable. The identifiability of the marginal distribution of $ \mathbf A^{(d+1)} $ (i.e., $ \boldsymbol {\pi }^{(d+1)} $ ) allows it to be treated theoretically as if it were observed when examining the identifiability for $ \mathbf Q^{(d+2)} $ , $ \boldsymbol {\beta }^{(d+2)} $ , and the marginal distribution of $ \mathbf A^{(d+2)} $ . Starting from the observed data layer ( $ d = 0 $ ) and proceeding one layer at a time, the identifiability of DeepCDMs can thus be inductively established.

This layer-wise identifiability rationale supports the procedure of generating samples of $ \mathbf A^{(d+1)} $ when estimating parameters of the $ d $ -th layer, for each d. Specifically, the identifiability of the model at layer $ d $ , as shown in Equation (24), implies that the distribution $ \mathbb {P}(\mathbf A^{(d+1)}) $ , from which samples of $ \mathbf A^{(d+1)} $ are drawn, can be uniquely identified. This theoretical guarantee justifies treating the sampled $ \mathbf A^{(d+1)} $ as observed data in the EM algorithm. The same procedure is applied recursively to the remaining $ D-1 $ layers.

3.6 Extending to confirmatory DeepCDMs

The layer-wise EM algorithm can be easily modified to fit confirmatory DeepCDMs, which assume all $\mathbf {Q}$ -matrices are known. In this case, each M-step solves the following optimization:

(25) $$ \begin{align} \boldsymbol{\beta}^{(d,t)} &= \underset{\boldsymbol{\beta}^{(d)}}{\text{arg max}} \,\widetilde{Q}^{(d,t)}, \quad d=1,\ldots,D. \end{align} $$

Compared to Equation (16), the terms $N \cdot P_{\boldsymbol {s}}(\boldsymbol {\beta }^{(d)})$ are dropped, as all the $\mathbf {Q}$ -matrices are known and do not need to be estimated. This makes the confirmatory case simpler than the exploratory case we have focused on. The layer-wise algorithm in Algorithm 1 can be readily used for parameter estimation, incorporating the known $\mathbf {Q}$ -matrices during the implementation of coordinate descent in the M-step.

The adaptation of the layer-wise EM algorithm for the confirmatory case is also a novel contribution, representing the first EM-type algorithm for confirmatory DeepCDMs. Unlike the Bayesian approach in Gu (Reference Gu2024), which fits models using an MCMC algorithm that can involve slower convergence and the need for prior specification, the layer-wise EM algorithm offers a computationally more efficient alternative by directly seeking the maximum likelihood estimator. We note that the layer-wise EM method could also be extended to incorporate prior distributions for computing maximum posterior estimates, should rich prior information be available, following the approach outlined in Gu (Reference Gu2024).

4.1 Simulation studies for main-effect DeepCDMs

Throughout this section, let $k_{d-1}$ and $k_d$ be the indices for the $(d{-}1)$ -th and d-th layers, respectively. For the main-effect models, the coefficients $\beta ^{(d)}_{k_{d-1},k_{d}}$ are specified as

(27) $$ \begin{align}\beta^{(d)}_{k_{d-1},0} = c^{(d)}_0, \quad \beta^{(d)}_{k_{d-1},k_{d}} = \frac{c^{(d)}_1}{\sum_{k_{d}=1}^{K_d} q_{k_{d-1},k_{d}}^{(d)}}, \quad \forall\, k_{d} \in [K_{d}],\ d=1,2,3, \end{align} $$

where $K_0 = J$ , and the constants $(c^{(d)}_0, c^{(d)}_1)$ are set to $(6, -3)$ , $(3, -1.5)$ , and $(3, -1.5)$ for $d=1, 2, 3$ , respectively. The RMSE and aBias are reported in Table 1. All indices decrease as the sample size increases, indicating improved estimation accuracy. For each sample size, the estimation accuracy is lower in the deeper layers than in the shallower layers, as reflected by the values of $\boldsymbol {\pi }^{(d)}$ and $P_{CR}^{(d)}$ . This result is intuitively reasonable, as estimating deeper layers is fundamentally more challenging due to stochastic latent layers. In addition, all RMSE and aBias values remain reasonably small, providing empirical support for the identifiability of the model.

Table 1

RMSE and aBias for the main-effect DeepCDM

To examine the recovery of the $\mathbf {Q}$ -matrices, we report the proportion of correctly estimated rows and entries in each $\mathbf {Q}^{(d)}$ for $d = 1, 2, 3$ , as shown in Table 2. Note that these indices are not directly comparable across layers at each sample size, as the proportions depend on both the size of the $\mathbf {Q}$ -matrix and the difficulty of parameter estimation. Due to the shrinkage ladder structure, which is supported by the identifiability conditions, the shallower layers contain larger $\mathbf {Q}$ -matrices, making their structures more challenging to recover. However, these layers also benefit from more informative signals, as they are closer to the observed data layer. In contrast, the deeper layers involve smaller $\mathbf {Q}$ -matrices that are structurally easier to recover, but they suffer from less informative signals due to the greater amount of uncertainty introduced at deeper levels. Despite these differences, when comparing $\mathbf {Q}$ -matrix recovery across sample sizes, it is evident that for each layer d, the estimation accuracy of $\mathbf {Q}^{(d)}$ improves as the sample size increases.

Table 2

Proportion of correctly recovered rows ( $P_{\text {Row-wise}}$ ) and entries ( $P_{\text {Entry-wise}}$ ) for the main-effect DeepCDM

4.2 Simulation studies for all-effect DeepCDMs

Denote the true coefficients as $\beta ^{(d)}_{k_{d-1},S_d}$ , $\forall S_d \subseteq [K_d] \backslash \emptyset $ and $\beta ^{(d)}_{k_{d-1},0} = c^{(d)}_0$ . In the all-effect DeepCDMs case, denote $\mathcal {K}_{k_{d-1}} = \left \{k_{d} \in [K_{d}];\, q_{k_{d-1},k_{d}}^{(d)} = 1\right \}$ , and $K_0 = J$ , and the first layer ( $d=1$ ) is modeled using the all-effect model with the parameters given below:

$$\begin{align*}\beta^{(d)}_{k_{d-1},S_d}=\left\{ \begin{array}{rcl} \frac{c_1^{(d)}}{2^{|\mathcal{K}_{k_{d-1}}|}} & & \prod_{l \in S_d} q_{k_{d-1},l}^{(d)} = 1 \\ 0 & & \prod_{l \in S_d} q_{k_{d-1},l}^{(d)} = 0. \\ \end{array} \right. \end{align*}$$

The two deeper layers ( $d=2,3$ ) are then modeled by the main-effect model, with parameters specified according to Equation (27). The constants $(c^{(d)}_0, c^{(d)}_1)$ are specified as $(c^{(1)}_0, c^{(1)}_1) = (6, -3)$ , $(c^{(2)}_0, c^{(2)}_1) = (3, -1.5)$ , and $(c^{(3)}_0, c^{(3)}_1) = (3, -1.5)$ . The $\mathbf Q^{(1)}$ -matrix is recovered by identifying, for each item, the highest-order nonzero interaction coefficient.

The RMSE and aBias values are summarized in Table 3. As expected, all indices decrease with increasing sample sizes, indicating improved estimation accuracy. RMSE and aBias values remain reasonably small across all layers, providing empirical support for the identifiability of the model. To evaluate the recovery of the $\mathbf {Q}$ -matrices, we report the proportions of correctly estimated rows and entries in each $\mathbf {Q}^{(d)}$ for $d = 1, 2, 3$ , as shown in Table 4. As in previous cases, these proportions are not directly comparable across layers, as they are influenced by differences in $\mathbf {Q}$ -matrix size and estimation difficulty. Nevertheless, for each layer d, the estimation accuracy of $\mathbf {Q}^{(d)}$ improves as the sample size increases.

Table 3

RMSE and aBias for the all-effect DeepCDM

Table 4

Proportion of correctly recovered rows ( $P_{\text {Row-wise}}$ ) and entries ( $P_{\text {Entry-wise}}$ ) for the all-effect DeepCDM

4.3 Simulation studies for DINA-effect DeepCDMs

Denote the true coefficients as $\beta ^{(d)}_{k_{d-1},S_d}$ for all $S_d \subseteq [K_d] \setminus \emptyset $ , with $\beta ^{(d)}_{k_{d-1},0} = c^{(d)}_0$ . In DINA DeepCDMs, the first layer ( $d=1$ ) is modeled using the DINA formulation where $\beta _{k_{d-1},S_d} = c_1^{(d)}$ if $S_d = \mathcal {K}_{k_{d-1}}$ , and zero otherwise, with $\mathcal {K}_{k_{d-1}}$ defined in Section 4.2. The DINA model can be viewed as a special case of the all-effect model, in which non-zero coefficients are assigned only to the interaction of all attributes required by the $(d{-}1)$ -th unit in each d-th layer model. This formulation allows the same estimation framework used for the all-effect model to be applied to the DINA model. The two deeper layers ( $d = 2, 3$ ) are modeled using the main-effect specification, with parameters defined as in Equation (27). The constants $(c^{(d)}_0, c^{(d)}_1)$ are specified as $(6, -3)$ , $(3, -1.5)$ , and $(3, -1.5)$ for $d = 1, 2, 3$ , respectively.

The RMSE and aBias results are reported in Table 5. Again, these values exhibit a clear decreasing trend with increasing sample size, indicating improved estimation accuracy. To assess the recovery of the $\mathbf {Q}$ -matrices, Table 6 presents the proportions of correctly estimated rows and entries for each $\mathbf {Q}^{(d)}$ , $d = 1, 2, 3$ . Overall, for all layers, the estimation accuracy of $\mathbf {Q}^{(d)}$ improves as the sample size increases.

Table 5

RMSE and aBias for the DINA DeepCDM

Table 6

Proportion of correctly recovered rows ( $P_{\text {Row-wise}}$ ) and entries ( $P_{\text {Entry-wise}}$ ) for the DINA DeepCDM