# Bayesian Analysis: Feature Adoption Forecast ## Question **Hypothesis**: New sharing feature will achieve >20% adoption within 3 months of launch **Estimating**: P(adoption >20%) **Timeframe**: 3 months post-launch (results measured at month 3) **Matters because**: Need 20% adoption to justify ongoing development investment. Below 20%, we should sunset the feature and reallocate resources. --- ## Prior Belief (Before Evidence) ### Base Rate What's the general frequency of similar features achieving >20% adoption? - **Reference class**: Previous features we've launched in this product category - **Historical data**: - Last 8 features launched: 5 achieved >20% adoption (62.5%) - Industry benchmarks: Social sharing features average 15-25% adoption - Our product has higher engagement than average - **Base rate**: 60% ### Adjustments How is this case different from the base rate? - **Factor 1: Feature complexity** - This feature is simpler than average (+5%) - Previous successful features averaged 3 steps to use - This feature is 1-click sharing - Simpler features historically perform better - **Factor 2: Market timing** - Competitive pressure is high (-10%) - Two competitors launched similar features 6 months ago - Early adopters may have already switched to competitors - Late-to-market features typically see 15-20% lower adoption - **Factor 3: User research signals** - Strong user request (+10%) - Feature was #2 most requested in last user survey (450 responses) - 72% said they would use it "frequently" or "very frequently" - Strong stated intent typically correlates with 40-60% actual usage ### Prior Probability **P(H) = 65%** **Justification**: Starting from 60% base rate, adjusted upward for simplicity (+5%) and strong user signals (+10%), adjusted down for late market entry (-10%). Net effect: 65% prior confidence that adoption will exceed 20%. **Range if uncertain**: 55% to 75% (accounting for uncertainty in adjustment factors) --- ## Evidence **What was observed**: Beta test with 200 users showed 35% adoption (70 users actively used feature) **How diagnostic**: This is moderately to strongly diagnostic evidence. Beta tests often show higher engagement than production (selection bias), but 35% is meaningfully above our 20% threshold. The question is whether this beta performance predicts production performance. ### Likelihoods **P(E|H) = 75%** - Probability of seeing 35% beta adoption IF true production adoption will be >20% **Reasoning**: - If production adoption will be >20%, beta should show higher (beta users are early adopters) - Typical pattern: beta adoption is 1.5-2x production adoption for engaged features - If production will be 22%, beta would likely be 33-44% → 35% fits this well - If production will be 25%, beta would likely be 38-50% → 35% is on lower end but plausible - 75% accounts for variance and beta-to-production conversion uncertainty **P(E|¬H) = 15%** - Probability of seeing 35% beta adoption IF true production adoption will be ≤20% **Reasoning**: - If production adoption will be ≤20% (say, 15%), beta would typically be 22-30% - Seeing 35% beta when production will be ≤20% would require unusual beta-to-production drop - This could happen (beta selection bias, novelty effect wears off), but is uncommon - 15% reflects that this scenario is possible but unlikely **Likelihood Ratio = 75% / 15% = 5.0** **Interpretation**: Evidence is moderately strong. A 35% beta result is 5 times more likely if production adoption will exceed 20% than if it won't. This is meaningful but not overwhelming evidence. --- ## Bayesian Update ### Calculation **Using odds form** (simpler for this case): ``` Prior Odds = P(H) / P(¬H) = 65% / 35% = 1.86 Likelihood Ratio = 5.0 Posterior Odds = Prior Odds × LR = 1.86 × 5.0 = 9.3 Posterior Probability = Posterior Odds / (1 + Posterior Odds) = 9.3 / 10.3 = 90.3% ``` **Verification using probability form**: ``` P(E) = [P(E|H) × P(H)] + [P(E|¬H) × P(¬H)] P(E) = [75% × 65%] + [15% × 35%] P(E) = 48.75% + 5.25% = 54% P(H|E) = [P(E|H) × P(H)] / P(E) P(H|E) = [75% × 65%] / 54% P(H|E) = 48.75% / 54% = 90.3% ``` ### Posterior Probability **P(H|E) = 90%** ### Change in Belief - **Prior**: 65% - **Posterior**: 90% - **Change**: +25 percentage points - **Interpretation**: Evidence strongly supports hypothesis. Beta test results meaningfully increased confidence that production adoption will exceed 20%. --- ## Sensitivity Analysis **How sensitive is posterior to inputs?** ### If Prior was different: | Prior | Posterior | Note | |-------|-----------|------| | 50% | 83% | Even starting at coin-flip, evidence pushes to high confidence | | 75% | 94% | Higher prior → very high posterior | | 40% | 77% | Lower prior → still high confidence | **Finding**: Posterior is somewhat robust. Evidence is strong enough that even with priors ranging from 40-75%, posterior stays in 77-94% range. ### If P(E|H) was different: | P(E\|H) | LR | Posterior | Note | |---------|-----|-----------|------| | 60% | 4.0 | 87% | Less diagnostic evidence → still high confidence | | 85% | 5.67 | 92% | More diagnostic evidence → very high confidence | | 50% | 3.33 | 82% | Weaker evidence → moderate-high confidence | **Finding**: Posterior is moderately sensitive to P(E|H), but stays above 80% across plausible range. ### If P(E|¬H) was different: | P(E\|¬H) | LR | Posterior | Note | |----------|-----|-----------|------| | 25% | 3.0 | 84% | Less diagnostic → still high confidence | | 10% | 7.5 | 94% | More diagnostic → very high confidence | | 30% | 2.5 | 80% | Weak evidence → moderate confidence | **Finding**: Posterior is sensitive to P(E|¬H). If beta-to-production drop is common (higher P(E|¬H)), confidence decreases meaningfully. **Robustness**: Conclusion is **moderately robust**. Across reasonable input ranges, posterior stays above 77%, supporting launch decision. Most sensitive to assumption about beta-to-production conversion rates. --- ## Calibration Check **Am I overconfident?** - **Did I anchor on initial belief?** - No - prior (65%) was based on base rates, not arbitrary - Evidence substantially moved belief (+25pp) - Not stuck at starting point - **Did I ignore base rates?** - No - explicitly used historical feature adoption (60%) as starting point - Adjusted for known differences systematically - **Is my posterior extreme (>90% or <10%)?** - Yes - 90% is borderline extreme - **Check**: Is evidence truly that strong? - LR = 5.0 is moderately strong (not very strong) - Prior was already high (65%) - Combination pushes to 90% - **Concern**: May be slightly overconfident - **Adjustment**: Consider reporting as 85-90% range rather than point estimate - **Would an outside observer agree with my likelihoods?** - P(E|H) = 75%: Reasonable - beta users are engaged, expect higher than production - P(E|¬H) = 15%: Potentially optimistic - beta selection bias could be stronger - **Alternative**: If P(E|¬H) = 25%, posterior drops to 84% (more conservative) **Red flags**: - ✓ Posterior is not 100% or 0% - ✓ Update magnitude (25pp) matches evidence strength (LR=5.0) - ✓ Prior uses base rates - ⚠ Posterior is at upper end (90%) - consider uncertainty range **Calibration adjustment**: Report as 85-90% confidence range to account for uncertainty in likelihoods. --- ## Limitations & Assumptions **Key assumptions**: 1. **Beta users are representative of broader user base** - Assumption: Beta users are 1.5-2x more engaged than average - Risk: If beta users are much more engaged (3x), production adoption could be lower - Impact: Could invalidate high posterior 2. **No major bugs or UX issues in production** - Assumption: Production experience will match beta experience - Risk: Unforeseen technical issues could crater adoption - Impact: Would make evidence misleading 3. **Competitive landscape stays stable** - Assumption: No major competitor moves in next 3 months - Risk: Competitor could launch superior version - Impact: Could reduce adoption below 20% despite strong beta 4. **Beta sample size is sufficient (n=200)** - Assumption: 200 users is enough to estimate adoption - Confidence interval: 35% ± 6.6% at 95% CI - Impact: True beta adoption could be 28-42%, adding uncertainty **What could invalidate this analysis**: - **Major product changes**: If we significantly alter the feature post-beta, beta results become less predictive - **Different user segment**: If we launch to a different user segment than beta testers, adoption patterns may differ - **Seasonal effects**: If beta ran during high-engagement season and launch is during low season - **Discovery/onboarding issues**: If users don't discover the feature in production (beta users were explicitly invited) **Uncertainty**: - **Most uncertain about**: P(E|¬H) = 15% - How often do features with ≤20% production adoption show 35% beta adoption? - This is the key assumption - If this is actually 25-30%, posterior drops to 80-84% - Recommendation: Review historical beta-to-production conversion data - **Could be wrong if**: - Beta users are much more engaged than typical users (>2x multiplier) - Novelty effect in beta wears off quickly in production - Production launch has poor discoverability/onboarding --- ## Decision Implications **Given posterior of 90% (range: 85-90%)**: **Recommended action**: **Proceed with launch** with monitoring plan **Rationale**: - 90% confidence exceeds decision threshold for feature launches - Even conservative estimate (85%) supports launch - Risk of failure (<20% adoption) is only 10-15% - Cost of being wrong: Wasted 3 months of development effort - Cost of not launching: Missing potential high-adoption feature **If decision threshold is**: - **High confidence needed (>80%)**: ✅ **LAUNCH** - Exceeds threshold, proceed with production rollout - **Medium confidence (>60%)**: ✅ **LAUNCH** - Well above threshold, strong conviction - **Low bar (>40%)**: ✅ **LAUNCH** - Far exceeds minimum threshold **Monitoring plan** (to validate forecast): 1. **Week 1**: Check if adoption is on track for >6% (20% / 3 months, assuming linear growth) - If <4%: Red flag, investigate onboarding/discovery issues - If >8%: Exceeding expectations, validate data quality 2. **Month 1**: Check if adoption is trending toward >10% - If <7%: Update forecast downward, consider intervention - If >13%: Exceeding expectations, high confidence 3. **Month 3**: Measure final adoption - If <20%: Analyze what went wrong, calibrate future forecasts - If >20%: Validate forecast accuracy, update priors for future features **Next evidence to gather**: - **Historical beta-to-production conversion rates**: Review last 5-10 feature launches to calibrate P(E|¬H) more accurately - **User segment analysis**: Compare beta user demographics to production user base - **Competitive feature adoption**: Check competitors' sharing feature adoption rates - **Early production data**: After 1 week of production, use actual adoption data for next Bayesian update **What would change our mind**: - **Week 1 adoption <3%**: Would update posterior down to ~60%, trigger investigation - **Competitor launches superior feature**: Would need to recalculate with new competitive landscape - **Discovery of major beta sampling bias**: If beta users are 5x more engaged, would significantly reduce confidence --- ## Meta: Forecast Quality Assessment Using rubric from `rubric_bayesian_reasoning_calibration.json`: **Self-assessment**: - Prior Quality: 4/5 (good base rate usage, clear adjustments) - Likelihood Justification: 4/5 (clear reasoning, could use more empirical data) - Evidence Diagnosticity: 4/5 (LR=5.0 is moderately strong) - Calculation Correctness: 5/5 (verified with both odds and probability forms) - Calibration & Realism: 3/5 (posterior is 90%, borderline extreme, flagged for review) - Assumption Transparency: 4/5 (key assumptions stated clearly) - Base Rate Usage: 5/5 (explicit base rate from historical data) - Sensitivity Analysis: 4/5 (comprehensive sensitivity checks) - Interpretation Quality: 4/5 (clear decision implications with thresholds) - Avoidance of Common Errors: 4/5 (no prosecutor's fallacy, proper base rates) **Average: 4.1/5** - Meets "very good" threshold for medium-stakes decision **Decision**: Forecast is sufficiently rigorous for feature launch decision (medium stakes). Primary area for improvement: gather more data on beta-to-production conversion to refine P(E|¬H) estimate.