🎪 Bayesian Circus: Iterative Guessing Challenge 🎲
Can You Infer the True Population from Limited Data?
Welcome to Bayesian Circus, where your job is to estimate the true distribution of colors in two mystery jars (Jar A & Jar B) using probability, logic, and strategic guessing.
This game follows a strict time limit per round, and you will track not just your best estimate, but also a confidence interval (+/- range) for each color.
Your goal is to refine your estimates as accurately as possible while avoiding overcorrection.
🏆 Points are awarded based on accuracy and how well you adjust over time.
📌 Game Objective
Your goal is to estimate the percentage of each color in Jars A & B by progressively updating your guesses as sample data is revealed.
- Each round, a new sample jar and associated counts will be revealed to all participants.
- You must update your probability estimates for both Jars A & B within a strict time limit.
- You must also provide a confidence interval (+/- range) to show how certain you are about each estimate.
- Overcorrecting by more than the allowed percentage results in a penalty.
🧐 Think carefully! Each sample gives you a better picture of the whole, but you never get to see the entire population! Your job is to make smart, measured updates as you receive more information.
📝 How Confidence Intervals Work
For each color estimate, you must include a confidence interval—a range that reflects how sure you are about your estimate.
- A wide range (e.g., 30% ±15%) means you are unsure and allowing for more error.
- A small range (e.g., 30% ±3%) means you are confident in your estimate.
- As you get more data, your confidence interval should narrow!
📝 Game Rules & Step-by-Step Guide
🔹 Step 1: Initial Guess (Prior Belief - No Data Yet)
- You will receive a Guessing Sheet listing Jar A and Jar B.
- Without any data, you must estimate the percentage of each color in both jars.
- You must also provide a confidence interval (+/- range) for each color, representing your uncertainty.
- Example Guesses:
- Jar A Guess: 30% Green (±10%), 25% Red (±5%), 15% Yellow (±15%)…
- Jar B Guess: 40% Yellow (±10%), 20% Red (±8%), 15% Green (±12%)…
- Example Guesses:
- You have 2 minutes to submit your guesses before the round closes.
🔹 Step 2: Small Sample (First Data Revealed)
- A pre-filled small sample is revealed for Jar A and Jar B.
- The exact counts for each color in the sample will be displayed for all participants.
- You must now revise your estimates for Jar A and Jar B based on the new data.
- Again, provide an updated confidence interval for each color.
- You have 2 minutes to submit your updated guesses.
🔹 Step 3: Additional Small Sample
- A second small sample is revealed for each jar.
- The exact counts for each color in the sample will be displayed for all participants.
- You update your estimates and confidence intervals based on this second sample.
- Overcorrection penalty applies:
- If your new estimate changes by more than 20% from the previous round, a penalty will be applied unless your final estimate is the closest to the true answer.
- You have 2 minutes to submit your updated guesses.
🔹 Step 4: Medium Sample
- A medium sample is revealed from each jar.
- Participants update their guesses again, refining both their estimates and confidence intervals.
- Overcorrection penalties still apply if estimates swing too much, but now the limit is 10%.
- You have 2 minutes to submit your updated guesses.
🔹 Step 5: Larger Data Release
- Three more medium sample jars are revealed for each population.
- Participants must now refine their guesses and narrow their confidence intervals as the data becomes clearer.
- Penalty for overcorrection still applies, and is now set at 5%
- 3 minutes are given for updates before the round closes.
🔹 Step 6: Final Round
- A large final sample is revealed for each population.
- Participants submit their final estimates and confidence intervals.
- Final scores are calculated.
🚨 Overcorrection Penalty System
🔺 If you change your estimate by more than the allowed percentage, you will receive a penalty unless your final guess is within 2% of the true proportion.
- Rounds 1-3: Max correction = 20%
- Round 4: Max correction = 10%
- Round 5: Max correction = 5%
This penalty encourages small, logical updates instead of wild swings!
💡 Strategy Tips for Participants
✔️ Avoid overcorrection! Bayesian updating is about small, rational adjustments.
✔️ Confidence intervals matter! If you’re consistently narrowing your range, you’re improving.
✔️ Pay attention to stabilization! If a color’s proportion isn’t changing much, it’s likely close to correct.
✔️ Use the medium and large samples wisely! They provide stronger data for accurate guessing.
📅 Game Flow
| Round | Action | Time Limit | Penalty for Overcorrection? |
|---|---|---|---|
| 1 | Initial Blind Guess | 2 min | ❌ No |
| 2 | Reveal Small Sample | 2 min | ✅ 20% |
| 3 | Reveal Another Small Sample | 2 min | ✅ 20% |
| 4 | Reveal Medium Sample | 2 min | ✅ 10% |
| 5 | Reveal Three More Medium Samples | 3 min | ✅ 5% |
| 6 | Reveal Final Large Sample | 3 min | ❌ No |
| 7 | Final Scoring & Leaderboard | - | - |
📋 Final Notes
🏆 Will your estimates converge on the truth, or will overcorrection lead you astray?
🎯 Think carefully, update wisely, and trust the data!
🎯 Bayesian Circus: Scoring Guide 📊
🔢 Scoring Formula
For each color in each round, your score is calculated as:
\[\text{Score} = \text{Base Points} \times \min\left(10, \frac{10}{\text{Confidence Interval Width}}\right)\]Key Rules
✔️ If the true value is inside your confidence interval, you earn points.
✔️ If the true value is NOT inside your confidence interval, you score 0 points for that color in that round.
✔️ Narrower confidence intervals earn higher scores!
✔️ A maximum multiplier of 10 applies to prevent extreme scores.
✔️ A minimum multiplier of 1 applies to prevent extreme scores.
🏆 Base Points Per Round
Each round, Base Points increase because later rounds have more data available.
| Round | Base Points (per color) | Max Possible Points per Color |
|---|---|---|
| 1 | 10 | 100 |
| 2 | 20 | 200 |
| 3 | 30 | 300 |
| 4 | 50 | 500 |
| 5 | 75 | 750 |
| 6 | 100 | 1000 |
📏 Confidence Interval Multiplier
Your confidence interval size affects your multiplier.
\[\text{Multiplier} = \min\left(10, \frac{10}{\text{Confidence Interval Width}}\right)\]- Smaller confidence intervals = higher scores!
- If the interval is too wide, you score fewer points.
- If the interval is too narrow and excludes the true value, you get 0 points.
- Confidence interval widths must be in whole numbers (e.g., ±0.5, ±1, ±1.5, etc.).
Example Multipliers
| Confidence Interval Width | Multiplier (10 / Width) | Applied Multiplier |
|---|---|---|
| 1.0% | 10 / 1 = 10 | 10 (max) |
| 2.0% | 10 / 2 = 5 | 5 |
| 5.0% | 10 / 5 = 2 | 2 |
| 10.0% | 10 / 10 = 1 | 1 |
| 20.0% | 10 / 20 = 0.5 | 1 (min) |
📊 Example Scoring Calculation
Example for One Color (e.g., Green in Jar A)
| Round | Base Points | Participant’s Confidence Interval | Width | Multiplier (10 / Width, max 10) | Score Earned |
|---|---|---|---|---|---|
| 1 | 10 | 10% - 50% | 40% | 1 (Min Multiplier) | 10 pts |
| 2 | 20 | 30% - 50% | 20% | 1 (Min Multiplier) | 20 pts |
| 3 | 30 | 35% - 45% | 10% | 1 (Min Multiplier) | 30 pts |
| 4 | 50 | 42% - 46% | 4% | 2.5 (10/4) | 125 pts |
| 5 | 75 | 43% - 45% | 2% | 5 (10/2) | 375 pts |
| 6 | 100 | 44% - 45% | 1% | 10 (Max) | 1000 pts |
🏆 Total Points for Green in Jar A: 1560 pts
💡 Key Takeaways
✔️ If the true percentage is inside your confidence interval, you earn points.
✔️ If the true percentage is outside your confidence interval, you score 0.
✔️ Smaller confidence intervals mean bigger scores!
✔️ Wider confidence intervals ensure safety but earn fewer points.
✔️ The later rounds are worth the most points!
🚨 Overcorrection Penalty
🔺 If you change your estimate by more than the allowed percentage, you lose 10% of the points for that round unless your guess for that round is within 2% of the true proportion.
| Round | Max Correction Allowed |
|---|---|
| 1-3 | 20% |
| 4 | 10% |
| 5 | 5% |
| 6 | No limit |
This discourages extreme jumps while still allowing necessary corrections as new data emerges.
📅 Game Flow
| Round | Action | Time Limit | Penalty for Overcorrection? |
|---|---|---|---|
| 1 | Initial Blind Guess | 2 min | ❌ No |
| 2 | Reveal Small Sample | 2 min | ✅ 20% |
| 3 | Reveal Another Small Sample | 2 min | ✅ 20% |
| 4 | Reveal Medium Sample | 2 min | ✅ 10% |
| 5 | Reveal Three More Medium Samples | 3 min | ✅ 5% |
| 6 | Reveal Final Large Sample | 3 min | ❌ No |
| 7 | Final Scoring & Leaderboard | - | - |
🏅 How the Winner is Determined
🏆 The participant with the highest total score across all 7 colors in both jars at the end of the game wins!
If two participants tie, the tiebreaker is:
- Who had the lowest average confidence interval width across all rounds.
- If still tied, who had the highest single-color score in any round.
📜 Final Summary
✅ All 7 colors are scored separately.
✅ Final score = sum of all color scores across all rounds.
✅ Smaller confidence intervals = higher score.
✅ Too wide an interval? You get fewer points. Too narrow? Risk getting 0.
✅ The later rounds are worth the most points.
✅ Overcorrection penalties apply to encourage rational updates.
🎯 Play smart, update wisely, and may the best Bayesian win! 🚀