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* Main BKT Computation Function
*
* This is the entry point for computing BKT state from problem history.
* It replays all SlotResults in chronological order to build up
* the current P(known) estimate for each skill.
*/
import { BKT_THRESHOLDS } from '../config/bkt-integration'
import type { ProblemResultWithContext } from '../session-planner'
import { calculateConfidence, getUncertaintyRange } from './confidence'
import { type BlameMethod, updateOnCorrect, updateOnIncorrectWithMethod } from './conjunctive-bkt'
import { helpWeight, responseTimeWeight } from './evidence-quality'
import { getDefaultParams } from './skill-priors'
import type {
BktComputeOptions,
BktComputeResult,
BktModeResult,
BktSkillState,
MasteryClassification,
SkillBktResult,
} from './types'
/** Extended options including blame method (not part of base BktComputeOptions to avoid breaking changes) */
export interface BktComputeExtendedOptions extends BktComputeOptions {
/** Which blame attribution method to use for incorrect multi-skill problems */
blameMethod?: BlameMethod
}
/**
* Default computation options.
*/
export const DEFAULT_BKT_OPTIONS: BktComputeExtendedOptions = {
confidenceThreshold: 0.5,
useCrossStudentPriors: false,
applyDecay: false,
decayHalfLifeDays: 30,
blameMethod: 'heuristic',
}
/**
* Apply time-based decay to P(known).
* Uses exponential decay toward the prior (pInit).
*
* @param pKnown - Current P(known) estimate
* @param daysSinceLastPractice - Days since the skill was practiced
* @param halfLifeDays - Half-life of decay in days
* @param pInit - Prior probability to decay toward
* @returns Decayed P(known)
*/
function applyTimeDecay(
pKnown: number,
daysSinceLastPractice: number,
halfLifeDays: number,
pInit: number
): number {
if (daysSinceLastPractice <= 0) return pKnown
// Exponential decay: after halfLifeDays, the "learned" portion decays by 50%
// pKnown decays toward pInit (prior), not toward 0
const decayFactor = 0.5 ** (daysSinceLastPractice / halfLifeDays)
const learnedPortion = pKnown - pInit
return pInit + learnedPortion * decayFactor
}
/**
* Compute BKT state for all skills from problem history.
*
* This is the main entry point - call it when displaying the Skills Dashboard.
* It replays all SlotResults in chronological order to compute the current
* P(known) estimate for each skill encountered.
*
* @param results - Problem results from session history
* @param options - Computation options (confidence threshold, blame method, etc.)
* @returns BKT results for all skills, sorted by need for intervention
*/
export function computeBktFromHistory(
results: ProblemResultWithContext[],
options: Partial<BktComputeExtendedOptions> = {}
): BktComputeResult {
// Merge with defaults so callers can override just what they need
const opts: BktComputeExtendedOptions = {
...DEFAULT_BKT_OPTIONS,
...options,
}
// Sort by timestamp to replay in chronological order
// Note: timestamp may be a Date or a string (from JSON serialization)
const sorted = [...results].sort((a, b) => {
const timeA =
a.timestamp instanceof Date ? a.timestamp.getTime() : new Date(a.timestamp).getTime()
const timeB =
b.timestamp instanceof Date ? b.timestamp.getTime() : new Date(b.timestamp).getTime()
return timeA - timeB
})
// Track state for each skill
const skillStates = new Map<string, BktSkillState>()
// Track per-mode skill states (mode → skillId → state)
const modeSkillStates = new Map<string, Map<string, BktSkillState>>()
// Process each problem result
for (const result of sorted) {
// Extract skill IDs from the result (skillsExercised tracks what was actually used)
const skillIds = result.skillsExercised ?? []
if (skillIds.length === 0) continue
// Check if this is a recency-refresh sentinel record
// These update lastPracticedAt but are ZERO-WEIGHT for BKT mastery (pKnown)
const isRecencyRefresh = result.source === 'recency-refresh'
// Ensure all skills have state initialized
for (const skillId of skillIds) {
if (!skillStates.has(skillId)) {
const params = getDefaultParams(skillId)
skillStates.set(skillId, {
pKnown: params.pInit,
opportunities: 0,
successCount: 0,
lastPracticedAt: null,
params,
})
}
}
// Initialize per-mode skill states if this result has a partType
const partType = (result as { partType?: string }).partType
if (partType) {
if (!modeSkillStates.has(partType)) {
modeSkillStates.set(partType, new Map())
}
const modeMap = modeSkillStates.get(partType)!
for (const skillId of skillIds) {
if (!modeMap.has(skillId)) {
const params = getDefaultParams(skillId)
modeMap.set(skillId, {
pKnown: params.pInit,
opportunities: 0,
successCount: 0,
lastPracticedAt: null,
params,
})
}
}
}
// For recency-refresh sentinels, only update lastPracticedAt - skip BKT calculation
if (isRecencyRefresh) {
const timestamp =
result.timestamp instanceof Date ? result.timestamp : new Date(result.timestamp)
for (const skillId of skillIds) {
const state = skillStates.get(skillId)!
// Update lastPracticedAt if this is more recent
if (!state.lastPracticedAt || timestamp > state.lastPracticedAt) {
state.lastPracticedAt = timestamp
}
// Also update mode-specific lastPracticedAt
if (partType) {
const modeState = modeSkillStates.get(partType)?.get(skillId)
if (modeState && (!modeState.lastPracticedAt || timestamp > modeState.lastPracticedAt)) {
modeState.lastPracticedAt = timestamp
}
}
}
continue // Skip pKnown updates for sentinel records
}
// Build skill records for BKT update
const skillRecords = skillIds.map((skillId: string) => {
const state = skillStates.get(skillId)!
return {
skillId,
pKnown: state.pKnown,
params: state.params,
}
})
// Calculate evidence weight based on help usage, response time, and retry status
const helpW = helpWeight(result.hadHelp)
const rtWeight = responseTimeWeight(result.responseTimeMs, result.isCorrect)
// Apply mastery weight from retry system (1.0 for first attempt, 0.5 for first retry, 0.25 for second retry)
// If masteryWeight is undefined (old data), default to 1.0
const retryWeight = result.masteryWeight ?? 1.0
const evidenceWeight = helpW * rtWeight * retryWeight
// Compute BKT updates (conjunctive model)
const blameMethod = opts.blameMethod ?? 'heuristic'
const updates = result.isCorrect
? updateOnCorrect(skillRecords)
: updateOnIncorrectWithMethod(skillRecords, blameMethod)
// Check if any update produced NaN (indicates invalid input data)
const hasInvalidUpdate = updates.some((u) => !Number.isFinite(u.updatedPKnown))
if (hasInvalidUpdate) {
console.warn(
'[BKT] Skipping problem result due to invalid data - skills:',
skillIds.join(', '),
'timestamp:',
result.timestamp,
'Updates with NaN:',
updates.filter((u) => !Number.isFinite(u.updatedPKnown)).map((u) => u.skillId)
)
continue // Skip this problem, don't corrupt skill states
}
// Apply updates with evidence weighting
for (const update of updates) {
const state = skillStates.get(update.skillId)!
// Weighted blend between old and new pKnown based on evidence quality
// When evidenceWeight = 1.0, use full update
// When evidenceWeight < 1.0, stay closer to prior
const newPKnown = state.pKnown * (1 - evidenceWeight) + update.updatedPKnown * evidenceWeight
// Final safety check - if blending still produces NaN, preserve prior state
if (!Number.isFinite(newPKnown)) {
console.warn(
'[BKT] Evidence blending produced NaN for skill:',
update.skillId,
'evidenceWeight:',
evidenceWeight,
'updatedPKnown:',
update.updatedPKnown,
'- preserving prior state'
)
continue
}
state.pKnown = newPKnown
state.opportunities += 1
if (result.isCorrect) state.successCount += 1
state.lastPracticedAt =
result.timestamp instanceof Date ? result.timestamp : new Date(result.timestamp)
}
// Per-mode BKT updates (independent computation using mode-specific priors)
if (partType) {
const modeMap = modeSkillStates.get(partType)!
const modeSkillRecords = skillIds.map((skillId: string) => {
const modeState = modeMap.get(skillId)!
return {
skillId,
pKnown: modeState.pKnown,
params: modeState.params,
}
})
const modeUpdates = result.isCorrect
? updateOnCorrect(modeSkillRecords)
: updateOnIncorrectWithMethod(modeSkillRecords, blameMethod)
const hasModeInvalidUpdate = modeUpdates.some((u) => !Number.isFinite(u.updatedPKnown))
if (!hasModeInvalidUpdate) {
for (const update of modeUpdates) {
const modeState = modeMap.get(update.skillId)!
const newPKnown =
modeState.pKnown * (1 - evidenceWeight) + update.updatedPKnown * evidenceWeight
if (!Number.isFinite(newPKnown)) continue
modeState.pKnown = newPKnown
modeState.opportunities += 1
if (result.isCorrect) modeState.successCount += 1
modeState.lastPracticedAt =
result.timestamp instanceof Date ? result.timestamp : new Date(result.timestamp)
}
}
}
}
// Convert overall states to results
const now = new Date()
const overall = convertStatesToResults(skillStates, opts, now)
// Build per-mode results
const byMode: Partial<Record<string, BktModeResult>> = {}
for (const [mode, modeMap] of modeSkillStates) {
if (modeMap.size > 0) {
byMode[mode] = convertStatesToResults(modeMap, opts, now)
}
}
return {
...overall,
byMode: Object.keys(byMode).length > 0 ? byMode : undefined,
}
}
/**
* Convert a map of skill states into sorted BKT results with classifications.
* Reused for both overall and per-mode result computation.
*/
function convertStatesToResults(
states: Map<string, BktSkillState>,
opts: BktComputeExtendedOptions,
now: Date
): { skills: SkillBktResult[]; interventionNeeded: SkillBktResult[]; strengths: SkillBktResult[] } {
const skills: SkillBktResult[] = []
for (const [skillId, state] of states) {
const successRate = state.opportunities > 0 ? state.successCount / state.opportunities : 0.5
const confidence = calculateConfidence(state.opportunities, successRate)
// Apply decay if enabled
let finalPKnown = state.pKnown
if (opts.applyDecay && state.lastPracticedAt) {
const daysSinceLastPractice =
(now.getTime() - state.lastPracticedAt.getTime()) / (1000 * 60 * 60 * 24)
finalPKnown = applyTimeDecay(
state.pKnown,
daysSinceLastPractice,
opts.decayHalfLifeDays,
state.params.pInit
)
}
const uncertaintyRange = getUncertaintyRange(finalPKnown, confidence)
// Classify mastery based on P(known) and confidence
const masteryClassification = classifyMastery(finalPKnown, confidence, opts.confidenceThreshold)
// Final safety check - this should not happen after skipping invalid updates
if (!Number.isFinite(finalPKnown)) {
console.warn(
`[BKT] UNEXPECTED: Skill ${skillId} has corrupted pKnown after processing: ${finalPKnown}. ` +
`Opportunities: ${state.opportunities}, SuccessCount: ${state.successCount}. ` +
'This should not happen - invalid updates should have been skipped.'
)
}
skills.push({
skillId,
pKnown: finalPKnown,
confidence,
uncertaintyRange,
opportunities: state.opportunities,
successCount: state.successCount,
lastPracticedAt: state.lastPracticedAt,
masteryClassification,
})
}
// Sort by pKnown ascending (weak skills first)
skills.sort((a, b) => a.pKnown - b.pKnown)
// Identify intervention needed (low pKnown with sufficient confidence)
const interventionNeeded = skills.filter((s) => s.masteryClassification === 'weak')
// Identify strengths (high pKnown with sufficient confidence)
const strengths = skills.filter((s) => s.masteryClassification === 'strong')
return { skills, interventionNeeded, strengths }
}
/**
* Classify a skill's mastery level based on P(known) and confidence.
*
* Uses unified thresholds from BKT_THRESHOLDS:
* - strong: P(known) >= 0.8
* - weak: P(known) < 0.5
* - developing: everything in between (or insufficient confidence)
*/
function classifyMastery(
pKnown: number,
confidence: number,
confidenceThreshold: number
): MasteryClassification {
// Need sufficient confidence to make strong claims
if (confidence < confidenceThreshold) {
return 'developing' // Not enough data to be sure either way
}
if (pKnown >= BKT_THRESHOLDS.strong) {
return 'strong'
} else if (pKnown < BKT_THRESHOLDS.weak) {
return 'weak'
} else {
return 'developing'
}
}
/**
* Recompute BKT with different options (e.g., different confidence threshold).
* Useful for UI controls that let users adjust the threshold.
*/
export function recomputeWithOptions(
bktResult: BktComputeResult,
newOptions: BktComputeOptions
): BktComputeResult {
// Re-classify all skills with new threshold
const skills = bktResult.skills.map((skill) => ({
...skill,
masteryClassification: classifyMastery(
skill.pKnown,
skill.confidence,
newOptions.confidenceThreshold
),
}))
const interventionNeeded = skills.filter((s) => s.masteryClassification === 'weak')
const strengths = skills.filter((s) => s.masteryClassification === 'strong')
return { skills, interventionNeeded, strengths }
}
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