V2.525 - Bayesian Model Showdown — Framework vs ΛCDM vs w₀wₐCDM
V2.525: Bayesian Model Showdown — Framework vs ΛCDM vs w₀wₐCDM
Status: NUANCED — Bayes factor favors framework over ΛCDM (+1.32); AIC/BIC favor ΛCDM; CMB is the discriminator
The Question
The framework predicts Ω_Λ = 149√π/384 = 0.6877 with zero free parameters. ΛCDM fits Ω_Λ ≈ 0.685 with one parameter. w₀wₐCDM uses 3 parameters. Which model does the combined data prefer when the Occam razor is properly applied?
Data
41 measurements from 5 datasets spanning z = 0 to z = 1090:
| Dataset | N | Source |
|---|---|---|
| BAO | 12 | DESI Y1 (7 redshifts, DM/DH correlations) |
| CMB | 3 | Planck 2018 compressed (R, l_a, Ω_b h²) |
| Cosmic chronometers | 24 | Moresco+2022 compilation |
| SNe Ia | 1 | Pantheon+ compressed (Ω_m = 0.334 ± 0.018) |
| BBN | 1 | Primordial D/H (Cooke+2018) |
Key Results
Chi-squared breakdown
| Dataset | Framework (0p) | Planck ΛCDM (1p) | Δχ² |
|---|---|---|---|
| BAO | 20.35 | 23.88 | -3.54 (framework wins) |
| CMB | 12.05 | 0.04 | +12.01 (ΛCDM wins) |
| CC | 13.50 | 13.70 | -0.20 |
| SNe | 1.46 | 1.08 | +0.38 |
| BBN | 0.26 | 0.26 | +0.00 |
| Total | 47.62 | 38.96 | +8.65 |
The framework’s total chi² is higher, driven almost entirely by the CMB compressed likelihood. The framework’s 0.44% higher Ω_Λ shifts the CMB shift parameter and acoustic scale enough to produce a 12-point chi² penalty.
Bayesian Evidence (Laplace approximation)
| Model | k | χ²_min | ln(Z) |
|---|---|---|---|
| Framework | 0 | 47.62 | -23.81 |
| ΛCDM | 1 | 38.73 | -25.13 |
| w₀wₐCDM | 3 | 25.72 | -23.56 |
Bayes Factors
| Comparison | ln(B) | Interpretation |
|---|---|---|
| Framework vs ΛCDM | +1.32 | Substantial evidence for framework |
| Framework vs w₀wₐCDM | -0.25 | Inconclusive |
| ΛCDM vs w₀wₐCDM | -1.57 | Substantial evidence for w₀wₐCDM |
Information Criteria
| Model | k | χ² | AIC | BIC |
|---|---|---|---|---|
| Framework | 0 | 47.62 | 47.62 | 47.62 |
| Best ΛCDM | 1 | 38.73 | 40.73 | 42.44 |
| Best w₀wₐ | 3 | 25.72 | 31.72 | 36.86 |
AIC and BIC both favor ΛCDM over the framework by ΔAIC = +6.9, ΔBIC = +5.2.
Interpretation
Why the Bayes factor and AIC/BIC disagree
The Bayes factor penalizes ΛCDM for its wasted prior volume: with a prior Ω_Λ ∈ [0.4, 0.9] but the data constraining Ω_Λ to ±0.001, ΛCDM “wastes” 99.8% of its prior. The framework pays no such penalty.
AIC/BIC penalize parameters at a fixed rate (2 per param for AIC, 3.7 per param for BIC), which is much less than the actual Bayesian Occam factor (~6 in log-evidence). This is because AIC/BIC are asymptotic approximations that assume the prior is uninformative relative to the likelihood — not true here.
The honest answer: both perspectives are legitimate.
- The Bayesian analysis says: “A theory that predicts Ω_Λ = 0.6877 with 0 parameters is more impressive than a theory that fits Ω_Λ = 0.685 with 1 parameter.”
- The frequentist analysis says: “The data prefer Ω_Λ = 0.685, and the framework’s value doesn’t fit as well.”
Where the tension lives
The entire Δχ² = 8.65 is dominated by the CMB compressed likelihood (Δχ² = +12.0). Specifically:
- The framework’s Ω_Λ = 0.6877 shifts the CMB acoustic scale l_a by 0.9 (vs Planck’s σ = 0.09)
- This 10σ pull in l_a drives the CMB chi²
The BAO data actually prefer the framework (Δχ² = -3.5), confirming V2.519.
The w₀wₐCDM question
The combined data show substantial evidence for w₀wₐCDM over ΛCDM (ln B = -1.57). The best-fit w₀ = -0.50, wₐ = -1.65 — both far from the cosmological constant (w₀ = -1, wₐ = 0). However, the framework and w₀wₐCDM are essentially tied (ln B = -0.25).
This confirms the DESI finding: there IS some pull toward w ≠ -1 in the combined data, driven by BAO. But the framework’s zero-parameter prediction is competitive with the 3-parameter w₀wₐCDM fit.
Critical Assessment
Strengths
- Bayes factor favors framework over ΛCDM: the Occam razor rewards zero parameters
- BAO data prefer the framework: Δχ² = -3.5 across 12 DESI measurements
- Framework vs w₀wₐCDM is a tie: 0 parameters matches 3-parameter fit
Weaknesses
- CMB compressed likelihood is the Achilles heel: 12-point chi² penalty from l_a mismatch
- Profile likelihood shows 5.6σ tension between framework’s Ω_Λ and the combined best-fit — this is serious
- Bayes factor is prior-dependent: a tighter ΛCDM prior (e.g., [0.6, 0.8]) would reduce the Occam penalty
- Pantheon+ constraint is compressed: full distance modulus analysis could shift results
- Cosmic chronometers have large uncertainties: contribute little constraining power
The fundamental tension
The framework predicts Ω_Λ = 0.6877. The CMB data (Planck) prefer 0.6847. The difference is only 0.44%, but the CMB constrains Ω_Λ so precisely that this generates significant tension in the combined fit.
This tension could be resolved by:
- Euclid/DESI Y5: if Ω_Λ moves from 0.6847 toward 0.688, the framework wins
- Systematic re-analysis: the CMB compressed likelihood may not capture the full Planck posterior
- The framework is wrong: if the trace anomaly prediction is off by 0.4%
Connection to Other Experiments
- V2.519: DESI BAO alone favors the framework (Δχ² = -3.2), confirmed here
- V2.520: Multi-probe stress test showed χ²/dof = 1.07, but used approximate methods
- V2.521: Forecasts 2.9σ combined discrimination by 2035 — this is driven by the CMB tension we quantify here
- V2.524: Euclid will distinguish n_grav = 10 vs 9 at 2.7σ, which would shift Ω_Λ enough to affect this comparison
Files
src/bayesian_showdown.py: Full cosmology engine, 5 datasets, evidence computationtests/test_bayesian_showdown.py: 32 tests (all passing)run_experiment.py: Complete 7-section analysisresults.json: Machine-readable results