Bayesian Analysis for Prediction Markets

import httpx

client = httpx.Client(base_url="https://api.rekko.ai/v1", headers={"Authorization": "Bearer YOUR_API_KEY"})
analysis = client.get("/markets/kalshi/KXFED-26MAR19/analysis", params={"expand": "causal"}).json()
for factor in analysis["causal"]["factors"]:
    print(f"  {factor['claim']}: {factor['prior']:.0%} → {factor['posterior']:.0%} (weight: {factor['weight']:.0%})")

What this page covers

Why Bayesian reasoning is well-suited to prediction markets
Prior estimation from base rates
Evidence gathering and likelihood updates
Causal factor decomposition
Automated Bayesian analysis via the Rekko API
Interpreting causal decomposition results

Why Bayesian reasoning for prediction markets?

Prediction markets price events as probabilities. The market price reflects the crowd’s aggregated estimate, but that estimate can be wrong — especially when:

New information has not been fully incorporated
The market is illiquid and slow to react
Participants have systematic biases (favorite-longshot bias, recency bias)

Bayesian reasoning provides a structured framework to form your own probability estimate by starting with a prior (base rate), updating with evidence, and arriving at a posterior probability you can compare against the market price.

The Bayesian framework

Step 1: Establish a prior

The prior is your starting estimate before looking at specific evidence. Good priors come from base rates:

Market question	Base rate source	Prior
Will the Fed cut rates?	Historical FOMC decisions	30% of meetings result in cuts
Will Bitcoin hit $150K?	Historical yearly BTC returns	Top-quartile years see 3x+ gains
Will inflation exceed 3%?	Historical CPI distribution	~15% of months since 2000

# Example: Fed rate decision
# Base rate: ~30% of FOMC meetings result in rate changes
prior = 0.30

Step 2: Gather evidence and update

For each piece of evidence, estimate how likely you would see that evidence if the event happens (likelihood) vs if it does not:

posterior = (prior × likelihood_yes) / (prior × likelihood_yes + (1-prior) × likelihood_no)

This is Bayes’ theorem applied to binary outcomes.

def bayesian_update(prior: float, likelihood_yes: float, likelihood_no: float) -> float:
    """Update a prior probability with new evidence."""
    numerator = prior * likelihood_yes
    denominator = numerator + (1 - prior) * likelihood_no
    return numerator / denominator

# Start with base rate
prob = 0.30

# Evidence 1: PCE inflation at 2.1% (within Fed target)
# If they will cut: 80% chance we'd see this data
# If they won't cut: 40% chance we'd see this data
prob = bayesian_update(prob, 0.80, 0.40)
print(f"After PCE data: {prob:.0%}")  # ~46%

# Evidence 2: Three FOMC members signal openness to cuts
# If they will cut: 90% chance of these signals
# If they won't cut: 20% chance
prob = bayesian_update(prob, 0.90, 0.20)
print(f"After FOMC signals: {prob:.0%}")  # ~79%

# Evidence 3: Strong employment report
# If they will cut: 30% chance of strong jobs (less likely)
# If they won't cut: 70% chance of strong jobs
prob = bayesian_update(prob, 0.30, 0.70)
print(f"After jobs report: {prob:.0%}")  # ~62%

Step 3: Compare with market price

Your posterior probability is your edge estimate:

market_price = 0.55  # Market says 55%
my_estimate = 0.62   # Your Bayesian posterior

edge = my_estimate - market_price  # +7 points
if edge > 0.05:  # Only trade with >5% edge
    print(f"BUY YES — edge: {edge:.0%}")

Causal factor decomposition

Instead of serial Bayesian updates, you can decompose the probability into weighted causal factors — independent claims that each push the probability in a direction. This approach:

Makes the analysis transparent and auditable
Identifies which factors matter most
Allows quick re-estimation when a single factor changes

Structure

Each causal factor has:

Claim: What the factor asserts
Direction: Does it support YES or NO?
Weight: How important is this factor relative to others (weights sum to ~1.0)
Confidence: How certain are you about this factor’s assessment?
Prior: Base probability before this factor’s evidence
Posterior: Updated probability after considering the evidence
Evidence: Specific data points supporting the assessment

Manual example

factors = [
    {
        "claim": "Inflation is within Fed's comfort zone",
        "direction": "supports_yes",
        "weight": 0.35,
        "confidence": 0.9,
        "prior": 0.50,
        "posterior": 0.78,
        "evidence": ["PCE Feb 2026: 2.1%", "Core CPI declining 3 months"],
    },
    {
        "claim": "Fed rhetoric is dovish",
        "direction": "supports_yes",
        "weight": 0.30,
        "confidence": 0.75,
        "prior": 0.50,
        "posterior": 0.68,
        "evidence": ["Waller speech March 12", "Bostic: 'open to adjustment'"],
    },
    {
        "claim": "Tariff uncertainty creates headwinds",
        "direction": "supports_no",
        "weight": 0.20,
        "confidence": 0.60,
        "prior": 0.50,
        "posterior": 0.42,
        "evidence": ["New tariffs announced March 5", "Trade deficit widening"],
    },
    {
        "claim": "Employment remains strong",
        "direction": "supports_no",
        "weight": 0.15,
        "confidence": 0.70,
        "prior": 0.50,
        "posterior": 0.38,
        "evidence": ["March NFP: +280K", "Unemployment: 3.8%"],
    },
]

# Weighted aggregation
overall = sum(f["weight"] * f["posterior"] for f in factors)
print(f"Overall probability: {overall:.0%}")  # ~63%

Automated causal decomposition with Rekko

The Rekko analysis API performs this decomposition automatically. Use ?expand=causal to get the full factor breakdown:

import httpx

client = httpx.Client(
    base_url="https://api.rekko.ai/v1",
    headers={"Authorization": "Bearer YOUR_API_KEY"},
    timeout=120.0,
)

# Get analysis with causal decomposition
analysis = client.get(
    "/markets/kalshi/KXFED-26MAR19/analysis",
    params={"expand": "causal"},
).json()

print(f"Overall probability: {analysis['probability']:.0%}")
print(f"Confidence: {analysis['confidence']:.0%}")
print(f"Edge vs market: {analysis['edge']:.1%}")
print()

causal = analysis["causal"]
print(f"Method: {causal['method']}")
print(f"Factors ({len(causal['factors'])}):")
for f in causal["factors"]:
    arrow = "↑" if f["direction"] == "supports_yes" else "↓"
    print(f"  {arrow} {f['claim']} (weight: {f['weight']:.0%}, conf: {f['confidence']:.0%})")
    print(f"    Prior: {f['prior']:.0%} → Posterior: {f['posterior']:.0%}")
    for e in f["evidence"]:
        print(f"    • {e}")

Example response

{
  "overall_probability": 0.71,
  "overall_confidence": 0.82,
  "method": "weighted_bayesian",
  "factors": [
    {
      "claim": "Inflation is within Fed's comfort zone",
      "direction": "supports_yes",
      "weight": 0.35,
      "confidence": 0.9,
      "prior": 0.6,
      "posterior": 0.82,
      "evidence": ["PCE Feb 2026: 2.1%", "Core CPI declining 3 months"]
    },
    {
      "claim": "Fed rhetoric is dovish",
      "direction": "supports_yes",
      "weight": 0.3,
      "confidence": 0.75,
      "prior": 0.5,
      "posterior": 0.68,
      "evidence": ["Waller speech March 12", "Bostic: 'open to adjustment'"]
    },
    {
      "claim": "Tariff uncertainty creates headwinds",
      "direction": "supports_no",
      "weight": 0.2,
      "confidence": 0.6,
      "prior": 0.4,
      "posterior": 0.45,
      "evidence": ["New tariffs announced March 5", "Trade deficit widening"]
    }
  ]
}

Aggregation methods

Method	Description
`weighted_bayesian`	Weighted average of factor posteriors (default)
`linear`	Simple weighted linear combination
`log_odds`	Aggregation in log-odds space (better for extreme probabilities)

Using causal decomposition in a trading bot

The causal structure is useful beyond a single analysis. You can:

Track factor changes over time — if the top-weighted factor shifts, re-analyze
Cross-reference factors across markets — the same “tariff uncertainty” factor appears in multiple markets
Build custom aggregation — weight factors differently based on your domain expertise

# Re-weight factors based on your own assessment
my_weights = {
    "Inflation is within Fed's comfort zone": 0.40,  # I weight this higher
    "Fed rhetoric is dovish": 0.25,
    "Tariff uncertainty creates headwinds": 0.25,  # I weight this higher too
}

my_prob = 0
for f in causal["factors"]:
    w = my_weights.get(f["claim"], f["weight"])
    my_prob += w * f["posterior"]

print(f"Rekko estimate: {analysis['probability']:.0%}")
print(f"My re-weighted estimate: {my_prob:.0%}")

What’s next

Causal decomposition

Full documentation of the causal factor schema.

Signals API

Trading signals that use Bayesian analysis for sizing.

Kelly criterion

Position sizing based on your probability estimate.

Getting Started

Core Concepts

Tutorials

Account

What this page covers

Why Bayesian reasoning for prediction markets?

The Bayesian framework

Step 1: Establish a prior

Step 2: Gather evidence and update

Step 3: Compare with market price

Causal factor decomposition

Structure

Manual example

Automated causal decomposition with Rekko

Example response

Aggregation methods

Using causal decomposition in a trading bot

What’s next

Causal decomposition

Signals API

Kelly criterion

Getting Started

Core Concepts

Tutorials

Account

Documentation Index

​What this page covers

​Why Bayesian reasoning for prediction markets?

​The Bayesian framework

​Step 1: Establish a prior

​Step 2: Gather evidence and update

​Step 3: Compare with market price

​Causal factor decomposition

​Structure

​Manual example

​Automated causal decomposition with Rekko

​Example response

​Aggregation methods

​Using causal decomposition in a trading bot

​What’s next

Causal decomposition

Signals API

Kelly criterion

What this page covers

Why Bayesian reasoning for prediction markets?

The Bayesian framework

Step 1: Establish a prior

Step 2: Gather evidence and update

Step 3: Compare with market price

Causal factor decomposition

Structure

Manual example

Automated causal decomposition with Rekko

Example response

Aggregation methods

Using causal decomposition in a trading bot

What’s next