HCIE vs BKT / DKT / SAKT / GKT

## HCIE — Mastery Update (Online Kalman Filter + Bayesian)// ── KALMAN FILTER per (learner i, concept k) ─────────────────────────// State: (μ_ik, σ²_ik)  — ability mean + variance// No offline training needed; updates on every attempt// Predict step (prior):μ̄_t  = μ_{t-1}              // no drift model in V1σ̄²_t = σ²_{t-1} + Q        // process noise Q = 0.01 (hyperparameter)// Update step (posterior):K_t  = σ̄²_t / (σ̄²_t + R)  // Kalman gain; R = obs noise ≈ 0.1μ_t  = μ̄_t + K_t · (r_t - μ̄_t)   // weighted move toward obsσ²_t = (1 - K_t) · σ̄²_t           // variance shrinks with each obs// complexity: O(1) — 5 scalar ops per attempt// space:      O(2) per (learner, concept) pair// ── BAYESIAN BETA POSTERIOR per (learner i, concept k) ────────────────// State: (α_ik, β_ik)  — Beta distribution over mastery prob// Init: α=1, β=1  (uniform prior)α_t = α_{t-1} + r_t          // add 1 on correctβ_t = β_{t-1} + (1 - r_t)   // add 1 on incorrectE[p_mastery] = α / (α + β)   // posterior meanVar[p_mastery] = α·β / ((α+β)²·(α+β+1))  // epistemic uncertainty// complexity: O(1) — 2 scalar increments// space:      O(2) per (learner, concept) pair// ── ENSEMBLE (Exponentiated Gradient) ─────────────────────────────────// 2 learners V2 (Kalman + Bayesian); initialised w = [0.5, 0.5]ŷ_t = w_Kalman · μ_t + w_Bayes · E[p_mastery]loss_l = (ŷ_l - r_t)²                    // per-learner squared lossw_l ← w_l · exp(-η · loss_l)  then /= Σ  // EG update, η=0.1// ── JT SCORE (recommendation, not prediction) ─────────────────────────// KEY DIFFERENCE: HCIE doesn't just predict — it scores next conceptsJT(i,k) = Σ_d weight_d · signal_d(i,k)// signals drive which concept to serve NEXT (policy, not estimator)

// Cold-start: seed state tiap konsep
1for c in C:  state(u,c) ← (m ← p̄(c) [def 0,3], σ² ← σ₀² [def 0,5])
2             repr_posterior(u,c) ← prior lemah
// --- REKOMENDASI (per permintaan) ---
3C_avail ← unlocked_concepts(u, C, dag, τ)      // gerbang prasyarat
4if C_avail = ∅:  C_avail ← foundation_concepts(C)
5for c in C_avail:
6    (m, σ²) ← read_posterior(u, c)
7    n ← 1/σ²;  α_c ← m·n;  β_c ← (1−m)·n     // ketidakpastian → kekuatan prior
8c* ← bandit.select_arm({Beta(α_c, β_c)}, context=π)   // Thompson: pilih konsep
9if c* invalid or c* ∉ C_avail:  c* ← fallback_concept(C_avail)
10(task, a*) ← pick_task(u, c*)                 // bandit representasi: pilih modalitas
11J_t(u,c*) ← σ(Σ wᵢ·Nᵢ) atas 6 dimensi          // pandangan tata kelola (ADC)
12emit RecommendationGenerated(u, c*, a*, J_t)  // event-sourcing
13serve (c*, a*)
// --- PEMBARUAN (saat hasil diamati) ---
14observe y ∈ {0,1} dan sinyal (waktu respons, dst.)
15(m, σ²) ← kalman_update(m, σ², y)             // Algoritma 3.1, O(1)
16repr_posterior(u, a*) ← beta_update(·, y)     // menutup lingkar bandit modalitas
17propagate_transfer(u, c*, dag)                // bukti → konsep turunan
18append_event(u, state sebelum/sesudah, aksi, sinyal)   // keterlacakan (audit)
19return (c*, a*), m

## Full loop: learner i attempts concept k, response r ∈ {0,1}// ─── 1. RECEIVE EVENT ─────────────────────────────────────────────────event = {learner_id: i, concept_id: k, correct: r, timestamp: t}// ─── 2. KALMAN UPDATE  O(1) ───────────────────────────────────────────state_ik = kf_store[i][k]          // (μ, σ²) loaded from DB / cacheK_t      = state_ik.var / (state_ik.var + R)state_ik.mean += K_t * (r - state_ik.mean)state_ik.var  *= (1 - K_t)// ─── 3. BAYESIAN UPDATE  O(1) ─────────────────────────────────────────beta_ik = beta_store[i][k]         // (α, β)beta_ik.alpha += rbeta_ik.beta  += (1 - r)// ─── 4. ENSEMBLE PREDICTION  O(learners=2) ────────────────────────────p_kalman = state_ik.meanp_bayes  = beta_ik.alpha / (beta_ik.alpha + beta_ik.beta)p_hat    = w_kalman * p_kalman + w_bayes * p_bayes// EG weight update omitted for brevity// ─── 5. NEXT-CONCEPT SELECTION — Thompson bandit  O(|C_avail|) ────────// Called when system needs to pick NEXT concept for learner iC_avail = prerequisite_gate(i, concept_graph, τ)     // DAG constrainsfor candidate k' in C_avail:    α,β   = beta_prior_from(μ_ik', σ²_ik')           // uncertainty → prior strength    score = thompson_contextual(α, β, zpd, ρ)        // Algoritma 3.6next_concept = softmax_sample(scores, τ=0.2)         // the BANDIT selectsJ_t = jt_compute(i, next_concept)                    // 6-dim governance VIEW → ADC                                                     // (recorded, does NOT select)// ─── 6. PERSIST ───────────────────────────────────────────────────────// experiment_trajectories: jt_delta_m_contribution, jt_challenge_contribution, ...// outbox_events → Kafka → projection-consumer → learner_projections// ADC reads trajectory columns post-hoc for governance audit// ─── TOTAL PER-EVENT COST ─────────────────────────────────────────────// Steps 1-4:  O(1)// Step 5:     O(|C_avail|) selection + O(deg(c)) J_t view ← only on//             recommendation request; skipped on pure prediction queries// No batch job needed, no model reload, works from first interaction

## AUC comparison (matched eval, sealed run — Tabel 4.8, tie-aware rank-AUC)// AUC = P(model ranks a correct attempt above an incorrect attempt)// = area under ROC curve// HCIE AUC=0.6051  — headline (canonical lagged m_K, single Kalman)// BKT  AUC=0.5963  — best classical baseline; EM-tuned per concept// DKT  AUC=0.5892  — underperforms on Junyi (small dataset; LSTM overfit risk)// SAKT AUC=0.5730  — attention needs longer sequences than Junyi provides// GKT  AUC=0.5711  — global node states hurt per-learner personalisation here// (the rejected 2-learner deployed readout — HCIE 0.609 vs BKT 0.612 —//  is disclosed on /review/baselines; it is NOT the headline)// COLD-START delta (≤5 attempts, pooled Junyi)://   HCIE - BKT = +0.037   HCIE leads in the sparse-data regime//   (reported as a lead, not significance — the CI crosses 0 at n=10)//   Mechanism: KF uncertainty + ZPD → explore before exploiting//              BKT needs several attempts before EM estimate converges//   As history grows (≤20, all): delta shrinks — BKT catches up//   This is the expected tradeoff: online priors help cold, EM helps warm// KEY CAVEAT: HCIE is solving a HARDER problem than DKT/SAKT/GKT//   Those models are pure predictors evaluated at test time.//   HCIE runs the full policy loop (recommend → observe → update) at eval time.//   It's closer to regret minimisation than AUC maximisation.

## Live decision path (deployed gateway, measured with k6)// p95 ≈ 24 ms per decision @ 40 req/s per IP — CPU only, no GPU, no batch queue// per interaction: O(K) scalar ops (constant rolling windows: 10/100/100/20)## Regenerating the thesis evidence (measured wall-clock)// 1) verify Tabel 4.8 from the sealed per-row CSV (10,500 rows × 5 models)://      sklearn roc_auc_score → 0.6051 / 0.5963 / 0.5892 / 0.5730 / 0.5711//      wall-clock ≈ 2 s            (re-measured 2026-07-04 — exact reproduction)// 2) full 5-model re-score from the DB (dump_matched_eval_perrow.py)://      > 10 min                    (DKT/SAKT torch inference on CPU dominates)// 3) full sealed-run replay through the live runtime (API + outbox + consumers)://      ≈ 7.7 events/s + ~90 s startup   (measured 2026-06-25)//      → anchor N=96,727 ≈ 3.5 h wall-clock## Training cost// HCIE: none — no offline training, no embeddings, 0 GPU-hours// DKT / SAKT / GKT: require offline training per dataset before any prediction//   (that asymmetry is the Pareto argument: competitive AUC at ~zero setup cost)