BLIM and Bayesian Assessment

The Basic Local Independence Model

The BLIM (Falmagne & Doignon, 2011, Ch. 11) is a probabilistic model for response patterns on a knowledge structure.

Parameters

For each item \(q\):

  • \(\beta_q\) (slip): \(P(\text{incorrect} \mid q \in K)\) — the probability of a careless error on a mastered item.

  • \(\eta_q\) (guess): \(P(\text{correct} \mid q \notin K)\) — the probability of a lucky guess on an unmastered item.

The condition \(\beta_q + \eta_q < 1\) is the informative-item condition: a mastered item is more likely to receive a correct response than an unmastered one. It rules out the response-flipping degeneracy of a single item; it does not by itself make the BLIM identifiable — identifiability depends on the knowledge structure and is checked with check_identifiability.

Likelihood

The probability of a response pattern \(R\) given a knowledge state \(K\):

\[P(R \mid K) = \prod_{q \in Q} P(R_q \mid K)\]

where:

\[\begin{split}P(R_q = 1 \mid K) = \begin{cases} 1 - \beta_q & \text{if } q \in K \\ \eta_q & \text{if } q \notin K \end{cases}\end{split}\]

The local independence assumption means responses to different items are conditionally independent given the state.

Bayesian assessment

Prior

The default is a uniform prior; a supplied state distribution may be used instead:

\[\pi_K^{(0)} = \frac{1}{|\mathcal{K}|}\]

Update

After observing a response \(R_q\) on item \(q\):

\[\pi_K^{(t+1)} \propto P(R_q \mid K) \cdot \pi_K^{(t)}\]

This is a standard Bayes update, applied sequentially for each observed response.

Item selection: Expected Information Gain

The EIG criterion selects the item that maximally reduces entropy:

\[\text{EIG}(q) = H(\pi^{(t)}) - \mathbb{E}_{R_q}\left[H(\pi^{(t+1)})\right]\]

where \(H\) is Shannon entropy and the expectation is over both possible responses, weighted by their marginal probability.

Items vs instances

In practice (e.g., ALEKS), each item is a problem type and has multiple instances — concrete questions of equivalent difficulty. Under the instance model, responses are assumed conditionally independent given the state and share item-level error parameters. The engine:

  1. Computes EIG at the item level.

  2. Selects a random un-asked instance of the best item.

  3. Updates the posterior on the parent item.

Equivalence is a modeling assumption to check for the actual instances; using the same parent item does not establish it empirically (Cosyn et al., 2021).

Parameter estimation (EM)

When \(\beta_q\), \(\eta_q\) are unknown, they can be estimated from data via the EM algorithm:

E-step: For each response pattern \(R_r\) and state \(K_k\):

\[w_{rk} = P(K_k \mid R_r) = \frac{P(R_r \mid K_k) \cdot \pi_k}{\sum_{k'} P(R_r \mid K_{k'}) \cdot \pi_{k'}}\]

M-step: For distinct patterns with observed frequencies \(f_r\), re-estimate parameters from frequency-weighted sufficient statistics:

\[\hat{\beta}_q = 1 - \frac{\sum_r \sum_{k: q \in K_k} f_r w_{rk} \cdot R_{rq}}{\sum_r \sum_{k: q \in K_k} f_r w_{rk}}\]
\[\hat{\eta}_q = \frac{\sum_r \sum_{k: q \notin K_k} f_r w_{rk} \cdot R_{rq}}{\sum_r \sum_{k: q \notin K_k} f_r w_{rk}}\]

The state probabilities update as

\[\hat{\pi}_k = \frac{\sum_r f_r w_{rk}}{\sum_r f_r}.\]

For respondent-level rows, \(f_r=1\). Exact EM cannot decrease the log-likelihood; finite precision and numerical clipping can introduce small deviations. Convergence does not guarantee the global maximum.

References

  • Falmagne, J.-C., & Doignon, J.-P. (2011). Learning Spaces, Ch. 11.

  • Cosyn, E., et al. (2021). A practical perspective on knowledge space theory: ALEKS and its data. Journal of Mathematical Psychology, 101.

  • Heller, J., & Wickelmaier, F. (2013). Minimum discrepancy estimation in probabilistic knowledge structures.