Bayes Factor Odds Ratio

---
name: bayes_factor_odds_ratio
description: Evaluates the likelihood of different binary black hole formation channels (CEE, CHE, SMT) given gravitational wave strain data by comparing Bayesian evidence and prior odds. Use when the user has predictions and gold and needs to compute Bayes factor ($\mathcal{B}$), Odds ratio ($\mathcal{O}$).
metadata:
  skill_kind: metric
  source_arxiv: 2211.05945
  bibtex_key: qin2022searching
  confidence: high
---

# bayes_factor_odds_ratio

> Searching for candidates of coalescing binary black holes formed through chemically homogeneous evolution in GWTC-3 — Ying Qin et al. (arXiv:2211.05945, 2022)

## What this evaluates

Evaluates the likelihood of different binary black hole formation channels (CEE, CHE, SMT) given gravitational wave strain data by comparing Bayesian evidence and prior odds.

## Datasets

- **GWTC-3** — total 90; splits: analysis (90)

## Metrics

- `Bayes factor ($\mathcal{B}$)` **(primary)** — range: other
  - Ratio of Bayesian evidences between two models (waveform + prior). Larger values indicate stronger support for the tested model over the reference (CEE).
- `Odds ratio ($\mathcal{O}$)` **(primary)** — range: other
  - $\mathcal{O} = \mathcal{B} \times (\mathcal{R}_i / \mathcal{R}_{\rm CEE})$, where $\mathcal{R}_i/\mathcal{R}_{\rm CEE}$ is the relative merger rate density of channel $i$ vs CEE. Values >3/30/100 indicate moderate/strong/very strong evidence.
- `Merger rate density ($\mathcal{R}0$)` — range: Gpc⁻³ yr⁻¹
  - Local ($z=0$) merger rate density derived per event and summed across channels, weighted by Bayes factors. Reported as 90% credible lower limits.

## Input / output format

**Input**: Gravitational wave strain data (17s duration), waveform model (IMRPhenomXP), power spectral density, and astrophysically motivated priors for chirp mass, mass ratio, and component spins.

**Output**: Posterior distributions for intrinsic parameters (chirp mass, mass ratio, spins), Bayes factors comparing formation channels, and odds ratios.

## Scoring recipe

```python
# Compute Bayes factor between CHE/SMT prior and CEE prior
B = evidence_CHE / evidence_CEE
# Compute odds ratio using relative merger rates from literature
R_CHE = 0.14  # relative fraction
R_CEE = 0.29
O = B * (R_CHE / R_CEE)
# Classify evidence strength
if O > 100: strength = "very strong"
elif O > 30: strength = "strong"
elif O > 3: strength = "moderate"
elif O > 1/3: strength = "comparable"
else: strength = "favors CEE"
```

## Common pitfalls

- Using 1D marginal distributions instead of the full 4D joint PDF for priors loses potential correlations between parameters.
- Assuming strictly aligned spins ($\theta_i=0$) ignores potential precessional effects, which may bias posterior estimates for some events.
- Prior choices strongly affect posterior distributions of mass ratio and spins, making results sensitive to model assumptions.

## Evidence (verbatim from paper)

> The Bayesian inference takes the strain data of an event, the waveform model, the power spectral density, and the prior for the parameters representing the binary’s properties as input, and returns the parameters’ posterior distributions as output. One can derive the Bayes factor, $\mathcal{B}$, between two models (waveform $+$ prior) by comparing the Bayesian evidence $\mathcal{E}$ *(Thrane & Talbot, [2019](#bib.bib89 ""); Zevin et al., [2020](#bib.bib102 ""))*. The Bayes factor reflects a comprehensive evaluation of the goodness-of-fit and the prior volume for the two models. Therefore, larger is the Bayes factor, more favored is the formation channel indicated.

## Citation

```bibtex
@misc{qin2022searching,
  title={Searching for candidates of coalescing binary black holes formed through chemically homogeneous evolution in GWTC-3},
  author={Ying Qin et al.},
  year={2022},
  note={arXiv:2211.05945}
}
```

- arXiv: 2211.05945