# 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.
