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