Bayesian modeling of low-energy nuclear reactions

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\section{Bayesian Inference and Model Discrimination}

% Replace the line below with a short description of what is being observed/measured
To quantitatively assess whether the observed [OBSERVABLE] is consistent with a [MECHANISM_1] or [MECHANISM_2] mechanism, we employ a Bayesian inference framework.

\subsection{Forward Model}

The observed [OBSERVABLE] is modeled as:

N_{\text{obs}}(t) = \frac{P t}{E_f}\,\theta \, \left(1 - e^{-t/\tau}\right) + \epsilon

where:
\begin{itemize}
\item P is the excess power,
\item E_f \approx [VALUE]\,\text{J} is the energy per [EVENT TYPE] event, % e.g. 3.8e-12 J for D+D fusion
\item \theta is a scaling factor relative to the ideal [MECHANISM_1] rate,
\item \tau is the [PRODUCT] leakage time constant, % e.g. helium leakage
\item \epsilon \sim \mathcal{N}(0,\sigma^2) represents measurement noise.
\end{itemize}

\subsection{Likelihood Function}

Assuming Gaussian measurement noise, the likelihood for a dataset \{N_i, t_i\} is:

\mathcal{L}(\theta,\tau,\sigma) = \prod_i \frac{1}{\sqrt{2\pi \sigma^2}} \exp\left( -\frac{ \left(N_i - N_{\text{model}}(t_i)\right)^2 }{2\sigma^2} \right)

\subsection{Prior Distributions}

We impose physically motivated priors:

\tau \sim \mathrm{LogNormal}(\ln \tau_0, \sigma_\tau), \quad \tau_0 \sim [TYPICAL VALUE] % e.g. 10 hr
\theta \sim \mathcal{N}(1, \sigma_\theta^2)
\sigma \sim \mathrm{HalfNormal}(\sigma_0)

These priors allow for moderate deviations from ideal [MECHANISM_1] behavior while enforcing physical plausibility.

\subsection{Posterior Distribution}

The posterior distribution is:

p(\theta,\tau,\sigma \mid \text{data}) \propto \mathcal{L}(\theta,\tau,\sigma) \, p(\theta)\,p(\tau)\,p(\sigma)

Posterior sampling (e.g., via MCMC) yields distributions for the model parameters.

\subsection{Confidence Intervals}

From posterior samples, credible intervals are defined as:

\mathrm{CI}_{95\%}(\theta) = [\theta_{2.5\%}, \theta_{97.5\%}]
\mathrm{CI}_{95\%}(\tau) = [\tau_{2.5\%}, \tau_{97.5\%}]

In practice, large intervals indicate weak parameter identifiability, reflecting experimental uncertainty.

\subsection{Model Comparison}

We compare two competing hypotheses:

\begin{itemize}
\item \textbf{H _0 : [MECHANISM_2]/Structural Model}

N_{\text{pred}} = 0

\item \textbf{H _1 : [MECHANISM_1] Model}

N_{\text{pred}}(t) = \frac{P t}{E_f}\,\theta \left(1 - e^{-t/\tau}\right)

\end{itemize}

The evidence for each model is:

p(\text{data} \mid H_j) = \int \mathcal{L}(\theta,\tau,\sigma)\, p_j(\theta,\tau,\sigma)\, d\theta\, d\tau\, d\sigma

\subsection{Bayes Factor}

The model comparison is quantified by the Bayes factor:

K = \frac{p(\text{data} \mid H_1)} {p(\text{data} \mid H_0)}

Interpretation:

\begin{center}
\begin{tabular}{ll}
\toprule
K & Interpretation \
\midrule
< 1 & [MECHANISM_2] model favored \
1–10 & Weak evidence for [MECHANISM_1] model \
10–100 & Moderate evidence \
> 100 & Strong evidence for [MECHANISM_1] model \
\bottomrule
\end{tabular}
\end{center}

\subsection{Implications}

If the [MECHANISM_1] model is correct, posterior samples should satisfy:

\theta \approx 1 \quad \text{and} \quad \tau \text{ finite and consistent across experiments}

Deviations (e.g., wide posteriors, inconsistent \theta ) indicate either:
\begin{itemize}
\item experimental uncertainty (leakage, detection limits), or
\item inconsistency with a [MECHANISM_1] mechanism.
\end{itemize}

\subsection{Key Result}

This framework establishes a quantitative, falsifiable test:

\boxed{ \text{A valid [MECHANISM_1] mechanism must simultaneously satisfy} }

\begin{itemize}
\item proportional heat–[PRODUCT] scaling,
\item consistent parameter inference ( \theta, \tau ),
\item statistically significant model preference (Bayes factor).
\end{itemize}

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