Fixed point analysis of the Knowledge Tracing Algorithm yields parameter constraints P(G)+P(S)<1 and 0<P(T)<(1−P(S))/(1−P(G)) for sensible behavior¶

Claim · All claims
Evidence · 1 study (2 entries) · theoretical r3 · q2
Currency · newest study 2013 · not yet checked against its sources · not reviewed by a person · overdue for a recheck since 2016: tool and model accuracy changes quickly

Subclaims¶

q2 i? For P(Lj|Oj) to remain in [0,1] and converge properly, the parameters must satisfy P(G)+P(S)<1 and 0<P(T)<(1−P(S))/(1−P(G)); the P(T) constraint completely supersedes the other. → Brett Van de Sande 2013 q2 i? With parameters consistent with negative learning (P(S)+P(G)>1), the algorithm's behavior inverts: P(Lj|Oj) decreases for correct steps and increases for incorrect steps. → Brett Van de Sande 2013 (2)

Evidence¶

Brett Van de Sande 2013¶

Van de Sande, B. (2013). Properties of the Bayesian Knowledge Tracing Model. Journal of Educational Data Mining, 5(2). doi:10.5281/zenodo.3554629

q2 · i? · theoretical · r3

Analytical fixed point analysis (Section 4) of the recursion relations Eqns. (11) and (12), illustrated in Fig. 3: stable fixed points at 1 and at (1−P(G))P(T)/(1−P(G)−P(S)) yield the constraints (15) and (16); Fig. 4 plots the allowed region.

"In order for P (Lj|Oj) to remain in the interval [0 , 1] for any starting value P (L0)∈ [0, 1] and any sequence of correct/incorrect steps Oj, we need the stable fixed point (14) to lie in the interval [0 , 1] and the unstable fixed point (13) to remain negative."

Brett Van de Sande 2013 (2)¶

Van de Sande, B. (2013). Properties of the Bayesian Knowledge Tracing Model. Journal of Educational Data Mining, 5(2). doi:10.5281/zenodo.3554629

q2 · i? · theoretical · r3

Analytical result (Section 4): under negative-learning parameters the recursion behavior inverts, so correct responses lower the knowledge estimate and incorrect responses raise it — the 'empirically degenerate' behavior Baker, Corbett, and Aleven defined.

"If we instead choose parameters consistent with negative learning,P (S)+P (G)> 1, we find that the behavior of P (Lj|Oj) becomes inverted: P (Lj|Oj) decreases for correct steps and increases for incorrect steps."

Discussion¶

Related Claims¶

• Under standard BKT with non-degenerate parameters, the mastery probability stays above the learn rate even after unboundedly many incorrect responses — related
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