Scope, assumptions and computational limits¶
The package implements finite, dichotomous Knowledge Space Theory, competence-based derivation, QUERY, ITA, three IITA variants with population evaluation and delta-method inference, BLIM/SLM estimation and Bayesian assessment. Completeness is stated by method and assumptions; there is no claim to implement every KST model or every function of another package.
Different kinds of limitation¶
Issue |
Where it comes from |
Consequence and available response |
|---|---|---|
Local independence and a fixed state during assessment |
BLIM and assessment model assumptions |
Check substantive plausibility; the current estimator does not test them automatically |
Nonidentifiable parameters |
The chosen structure and parameterization |
Inspect the constrained Jacobian; compare observable probabilities; justified fixed/shared parameters change the model and may change its rank |
Local optima and stopping tolerance |
EM and floating-point optimization |
Use restarts and inspect convergence; convergence does not certify the global optimum |
Many states |
Structure size and representation |
A powerset has exponentially many states, but a chain has only $ |
Reliability enumeration |
The implemented finite-sum algorithm |
Chunking bounds working arrays; all response patterns are still visited. This is not a proof that every reliability algorithm must enumerate |
Sparse-table asymptotics or boundary fits |
Statistical approximation and model regularity |
Read conventional GOF/AIC/BIC cautiously; bootstrap calibration also depends on the model and fitting procedure |
SLM prior product requires a learning space |
Hypothesis guaranteeing normalization for every solvability vector |
Arbitrary structures are rejected; their masses are not repaired by normalization |
Single-relation IITA null diff=0 has zero gradient |
Nonregular squared-discrepancy null |
First-order normal calibration is invalid; the Z-test returns a warning and no p-value or interval |
Relations selected on the same data |
Selection uncertainty |
IITA inference treats relations as fixed; no post-selection correction is implemented |
Incomplete binary responses |
Separate observed-data ML contract |
BLIM marginalizes NaN; ignorability needs assumptions. Fixed-mask bootstrap needs exogenous masks. MNAR, incomplete SLM/MD/IITA and polytomous estimators remain gaps |
Effective fringes restricted to conjunctive competence models |
Hypotheses of the implemented 2017 results |
Do not extend the theorem automatically to alternative competencies in |
Aggregate probability data |
The loader’s explicit selection and aggregation |
Frequencies cannot reconstruct pairing; use the separate individual loader, which retains source IDs and missingness |
No learning transitions or general process model |
Current package coverage |
A static posterior update is not a temporal learning model |
Structural conventions¶
KnowledgeStructure stores states explicitly and requires the empty state
and full domain. Union closure, accessibility, well-gradedness and
discriminativity are distinct properties. A general knowledge structure
must not be silently converted to its union closure to use a space-only
algorithm. KnowledgeBase.from_structure checks this requirement.
For a relation pair (a, b), mastery of b implies mastery of a.
For compatibility, SurmiseRelation stores the supplied pairs as
generators, adding reflexive pairs. Call .transitive_closure() when a
closed quasi-order is required. Structure construction takes this closure
internally. A quasi-order may have equivalent items; its quotient is a
partial order. The ordinal case additionally requires discriminativity.
Inner and outer fringes are structural boundaries. An inner-fringe item could be the last addition on a compatible path; a state alone does not identify the learner’s actual history. A hanging state is allowed in a general knowledge structure, even though it excludes a learning space.
Statistical conventions¶
Error parameters mean slip beta and guess eta, not success and error
in interchangeable order. Estimation permits reversed or uninformative
items, with diagnostics; the assessment model requires
beta + eta < 1. Sharing an initial value does not constrain the fitted
values to be equal: use BLIMConstraints for that hypothesis.
Reported npar counts free coordinates after fixed/equality constraints.
It is not automatically the dimension of the observable model. Jacobian
rank is numerical evidence with a reported tolerance; it does not provide
a symbolic or global identifiability proof. Fitting and bootstrap
refitting use the same constraints. Bootstrap replication counts and
convergence information accompany the Monte Carlo p-value.
Evidence and extension policy¶
Definition-based exhaustive checks on small domains, independent numerical oracles, cross-software comparisons, and formal mathematical results serve different purposes. Agreement with R verifies compatibility on specified cases; it does not by itself prove a theorem or universal correctness.
Future methods need a primary reference, stated assumptions, parameter conventions, boundary behavior, relevant tests and provenance. Priority extension areas include explicit nonignorable missingness models, incomplete SLM/MD/IITA, temporal learning models and larger structures. Each needs its own scientific specification before entering the stable API.