Subjective Probability

#Definition

Subjective probability is an individual's personal assessment of the likelihood of an event, based on their own information, analysis, and judgment. Unlike objective probability (derived from repeated experiments or mathematical certainty), subjective probability reflects personal belief and can differ between individuals and from market prices.

In prediction markets, subjective probability is the foundation of trading: you trade when your subjective probability differs from the market's implied probability, believing your assessment is more accurate.

#Why It Matters in Prediction Markets

Subjective probability is central to prediction market trading and market function:

Trading basis: Every trade reflects a disagreement between the trader's subjective probability and the market price. Without subjective probabilities, there would be no reason to trade.

Edge identification: Your edge exists when your subjective probability is more accurate than the market's implied probability. Recognizing and quantifying this difference is essential.

Information aggregation: Prediction markets work by combining many traders' subjective probabilities into a consensus price. Each trader's subjective estimate contributes information.

Self-assessment: Understanding your own subjective probability clearly—before looking at market prices—prevents anchoring and enables genuine independent judgment.

Calibration tracking: Comparing your subjective probabilities to outcomes over time reveals whether your assessments are well-calibrated or systematically biased.

#How It Works

#Subjective vs. Objective Probability

Objective probability: Derived from repeated trials or mathematical structure

• Coin flip: 50% (by definition)
• Drawing a red card from a standard deck: 50% (26/52)

Subjective probability: Personal assessment based on available information

• Candidate X wins election: "I believe 62%"
• Company Y announces layoffs this month: "I estimate 35%"

For unique, one-time events like elections, only subjective probability applies—there's no frequency to measure.

#Components of Subjective Assessment

Your subjective probability emerges from:

1. Base rates: Historical frequencies of similar events
2. Specific information: Unique factors for this particular event
3. Analysis: How you weight and combine information
4. Intuition: Pattern recognition from experience
5. Biases: Cognitive tendencies that may distort assessment

#Numerical Example: Election Assessment

Building a subjective probability:

Base rate analysis:

• Incumbents win 70% of elections in this context
• Starting point: 70%

Specific factors:

• Economy is weak (-10%): Adjust to 60%
• Incumbent has scandal (-5%): Adjust to 55%
• Challenger is weak (+8%): Adjust to 63%
• Recent polls show close race (-3%): Adjust to 60%

Subjective probability: 60% incumbent wins

Market price: $0.52

Implication: You believe incumbent is underpriced. If your assessment is more accurate, buying incumbent shares has positive expected value.

#Subjective Probability vs. Market Price

The size of the gap matters. A small difference may not justify trading after accounting for fees and uncertainty about your own assessment.

#Degrees of Confidence

Not all subjective probabilities carry equal confidence:

High confidence (tight range): "I estimate 55-60%"

• Strong information, good analysis
• Willing to trade significant size

Low confidence (wide range): "I estimate 40-70%"

• Limited information, uncertain analysis
• Trade small or not at all

Explicitly acknowledging uncertainty in your subjective probability improves decision-making.

#Measuring Accuracy

To evaluate your subjective probabilities, use the Brier Score, which measures the squared difference between your probability (P) and the actual outcome (O, which is 1 or 0).

def calculate_brier_score(predictions, outcomes):
    """
    Calculate Brier Score for a set of probabilistic predictions.
    Lower score is better (0 = perfect, 0.25 = random guessing at 50%).
    """
    squared_errors = []
    for p, o in zip(predictions, outcomes):
        # p is subjective probability (e.g., 0.70)
        # o is outcome (1 for happened, 0 for didn't)
        squared_errors.append((p - o) ** 2)
        
    return sum(squared_errors) / len(squared_errors)

# Example
preds = [0.8, 0.6, 0.9, 0.2]
results = [1, 0, 1, 0] # 2nd prediction was wrong (60% yes, but didnt happen)
score = calculate_brier_score(preds, results) 
# Result: 0.1025 (Excellent forecasting)

#Examples

Expert domain knowledge: A regulatory specialist follows a pending approval decision. Based on precedent, agency composition, and informal signals, they assess 65% approval probability. Market price is $0.50. Their specialized knowledge produces a subjective probability that differs from market consensus—a potential trading opportunity.

Information integration: A trader combines multiple information sources—polls, prediction models, ground reports, historical patterns—into a subjective probability of 58% for a political outcome. The market shows $0.62. Their integrated assessment is lower than market price, suggesting the market may be overpriced.

Intuition from experience: An experienced prediction market trader develops intuitions about how markets behave after certain events. They assess 70% probability that a market will revert after an overreaction to news. This pattern-based subjective probability guides their trading.

Overconfident subjective probability: A trader strongly believes a candidate will win (90% subjective probability) based on enthusiasm they observe in their social circle. Market price is $0.55. Their subjective probability reflects sampling bias—they're overweighting their immediate environment.

#Risks and Common Mistakes

Overconfidence: Subjective probabilities tend to be overconfident—too extreme and too certain. A probability that "feels like" 80% might warrant only 65% given actual uncertainty.

Anchoring on market price: Checking market price before forming your assessment contaminates your subjective probability. You anchor on the visible number rather than reasoning independently.

Information asymmetry blindness: Your subjective probability reflects your information. But the market reflects many traders' information. When you disagree with the market, consider whether others know something you don't.

Unstable assessments: If your subjective probability changes dramatically with minor new information, it wasn't well-founded. Genuine subjective probabilities should be somewhat stable.

Precision illusion: Stating "I believe 63.7% probability" implies precision that doesn't exist. Subjective probabilities should acknowledge their imprecision.

Confirmation bias in assessment: Weighting information that confirms your existing belief while discounting contrary information produces biased subjective probabilities.

#Practical Tips for Traders

• Form assessments before checking prices: Write down your subjective probability before looking at the market. This preserves independent judgment.

• Express ranges, not points: Instead of "I think 60%," try "I think 55-65%, with 60% as central estimate." This captures uncertainty.

• Identify your information sources: List what information supports your assessment. Can you articulate why you believe what you believe?

• Consider the outside view: Start with base rates before adjusting for specific factors. This counteracts overweighting of specific information.

• Track and calibrate: Record your subjective probabilities and compare to outcomes over time. Are your 70% estimates correct about 70% of the time?

• Quantify your edge: Before trading, explicitly calculate: "My subjective probability is X, market is Y, so my perceived edge is X-Y." Only trade if this edge justifies the risk.

• Update, don't abandon: When new information arrives, update your subjective probability using Bayesian reasoning. Don't abandon your prior assessment entirely unless evidence is overwhelming.

#Related Terms

• Expected Value (EV)
• Posterior Probability
• Conditional Probability
• Market Efficiency
• Price Discovery
• Wisdom of Crowds
• Information Aggregation

#FAQ

#What is subjective probability in simple terms?

Subjective probability is your personal estimate of how likely something is to happen. Unlike a coin flip where probability is mathematically defined, events like elections or policy decisions have no "true" probability we can measure. Your subjective probability is your best guess based on what you know and how you analyze it. Different people can have different subjective probabilities for the same event.

#How is subjective probability different from market price?

Market price reflects the aggregate of many traders' subjective probabilities, weighted by how much they trade. Your subjective probability is just yours—your personal assessment based on your information and analysis. When your subjective probability differs from market price, that's a potential trading opportunity (if you're right) or a signal that you might be missing something (if the market is right).

#Why do subjective probabilities differ between people?

Different subjective probabilities arise from: different information (you know something others don't, or vice versa), different analysis (you weight factors differently), different biases (cognitive tendencies affect assessment), and different expertise (domain knowledge shapes interpretation). These differences are what make markets work—disagreement drives trading.

#How do I know if my subjective probability is accurate?

Track your assessments over time. If your 70% predictions come true about 70% of the time, you're well-calibrated. If your 70% predictions come true 50% of the time, you're overconfident. Calibration tracking requires recording many predictions and comparing to outcomes—a practice called "calibration training."

#Should I always trade when my subjective probability differs from market price?

Not necessarily. Consider: (1) How confident are you in your assessment? (2) How large is the difference? (3) What trading costs apply? (4) Could the market know something you don't? Small differences, low confidence, or high costs may make trading unprofitable even when you disagree with the market. Only trade when your edge clearly exceeds your uncertainty and costs.