## Line Graph: Bayes Factor for G: θ=1 vs G^c: θ~Beta(1,1) ### Overview The image is a logarithmic-scale line graph comparing the Bayes Factor (BF) in favor of G=1 against G^c with a Beta(1,1) prior for θ. The x-axis represents the number of trials (n) on a logarithmic scale (10⁰ to 10⁶), while the y-axis shows the log₁₀(BF) in favor of G=1 (0 to 6). A yellow line represents the theoretical relationship log₁₀(BF) = log₁₀(n+1), and orange crosses mark reference values at specific n points. --- ### Components/Axes - **Title**: "Appendix B3 — Bayes Factor for G: θ=1 vs G^c: θ~Beta(1,1)" - **X-axis**: - Label: "n (number of trials)" - Scale: Logarithmic (10⁰, 10¹, 10², ..., 10⁶) - Ticks: 10⁰, 10¹, 10², 10³, 10⁴, 10⁵, 10⁶ - **Y-axis**: - Label: "log₁₀ Bayes Factor in favor of G=1" - Scale: Linear (0 to 6, increments of 1) - **Legend**: - Top-right corner - Yellow line: "log₁₀(BF) = log₁₀(n+1)" - Orange crosses: "Reference values" --- ### Detailed Analysis 1. **Yellow Line (log₁₀(BF) = log₁₀(n+1))**: - Starts at approximately (10⁰, 0.3) and increases steadily. - At n=10⁰: log₁₀(1+1) ≈ 0.3 - At n=10¹: log₁₀(10+1) ≈ 1.04 - At n=10²: log₁₀(100+1) ≈ 2.004 - At n=10³: log₁₀(1000+1) ≈ 3.0004 - At n=10⁴: log₁₀(10,000+1) ≈ 4.00004 - At n=10⁵: log₁₀(100,000+1) ≈ 5.000004 - At n=10⁶: log₁₀(1,000,000+1) ≈ 6.0000004 2. **Orange Crosses (Reference Values)**: - Placed at n=10⁰, 10¹, 10², 10³, 10⁴, 10⁵, 10⁶. - Corresponding y-values: 1, 2, 3, 4, 5, 6. - Example: At n=10³, the cross is at y=3, matching log₁₀(1000+1) ≈ 3.0004. --- ### Key Observations - The yellow line perfectly aligns with the orange crosses, confirming the theoretical relationship log₁₀(BF) = log₁₀(n+1). - The Bayes Factor grows logarithmically with n, indicating diminishing returns in evidence strength as trials increase. - No outliers or deviations between the line and reference points. --- ### Interpretation - **Bayesian Inference Context**: - The Beta(1,1) prior for G^c implies a uniform prior over θ, suggesting no initial bias toward G=1 or G^c. - The Bayes Factor (BF) quantifies evidence in favor of G=1 relative to G^c. A BF > 1 indicates increasing support for G=1 as n grows. - **Logarithmic Growth**: - The log₁₀(BF) = log₁₀(n+1) relationship implies that each additional trial contributes less to the BF than the previous, consistent with Bayesian updating under a uniform prior. - **Practical Implication**: - To achieve a BF of 6 (strong evidence), ~10⁶ trials are required. This highlights the computational or practical challenges of high-confidence Bayesian inference in this scenario. --- ### Spatial Grounding & Trend Verification - **Legend Placement**: Top-right corner, clearly separated from the plot. - **Line vs. Crosses**: The yellow line (theoretical) and orange crosses (reference) are visually aligned, confirming accuracy. - **Axis Scales**: Logarithmic x-axis ensures exponential growth in trials is represented linearly, while the linear y-axis emphasizes proportional changes in BF. --- ### Content Details - **Theoretical Line**: - Equation: log₁₀(BF) = log₁₀(n+1) - Derived from Bayesian updating rules for G=1 vs G^c with θ~Beta(1,1). - **Reference Points**: - Explicitly marked at n=10ᵏ (k=0 to 6), with y-values matching log₁₀(n+1). --- ### Final Notes - The graph demonstrates a clear, deterministic relationship between trials and evidence strength, with no noise or variability. - The Beta(1,1) prior’s role in shaping the BF’s growth is critical for interpreting the results in Bayesian hypothesis testing.