> ## Documentation Index > Fetch the complete documentation index at: https://rekko.ai/docs/llms.txt > Use this file to discover all available pages before exploring further. # Bayesian Analysis for Prediction Markets > How to apply Bayesian reasoning to prediction markets. Prior estimation, evidence updating, causal factor decomposition, and automated Bayesian analysis via API. ```python theme={null} 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 | ```python theme={null} # 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. ```python theme={null} 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: ```python theme={null} 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 ```python theme={null} 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: ```python Python theme={null} 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}") ``` ```bash cURL theme={null} curl "https://api.rekko.ai/v1/markets/kalshi/KXFED-26MAR19/analysis?expand=causal" \ -H "Authorization: Bearer YOUR_API_KEY" ``` ```javascript JavaScript theme={null} const resp = await fetch( "https://api.rekko.ai/v1/markets/kalshi/KXFED-26MAR19/analysis?expand=causal", { headers: { Authorization: "Bearer YOUR_API_KEY" } } ); const analysis = await resp.json(); console.log(`Probability: ${analysis.probability}`); analysis.causal.factors.forEach((f) => { console.log(` ${f.direction === "supports_yes" ? "↑" : "↓"} ${f.claim} (${f.weight})`); }); ``` ### Example response ```json theme={null} { "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: 1. **Track factor changes over time** — if the top-weighted factor shifts, re-analyze 2. **Cross-reference factors across markets** — the same "tariff uncertainty" factor appears in multiple markets 3. **Build custom aggregation** — weight factors differently based on your domain expertise ```python theme={null} # 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 Full documentation of the causal factor schema. Trading signals that use Bayesian analysis for sizing. Position sizing based on your probability estimate.