TEXAS Proxy System Model — Bayesian Inference Workflow

TEXAS Proxy System Model — Bayesian Inference Workflow#

1. Overview#

The TEXAS Proxy System Model (PSM) is designed to link environmental variables (e.g., sea surface temperature, archaeal community composition, nutrient availability) to proxy measurements (e.g., GDGT-based Ring Index, TEX₈₆) using a Bayesian framework.

It has two complementary components:

  1. Forward Model — Predicts proxy values from known environmental conditions.

  2. Inverse Model (Bayesian Inference) — Infers environmental conditions from observed proxy values.

2. Why Bayesian?#

Bayesian inference provides a natural way to:

  • Combine prior knowledge (from culture experiments, core-top datasets, or theory) with new data.

  • Quantify uncertainty in both model parameters and reconstructions.

  • Explicitly propagate uncertainties from forward models into reconstructions.

The heart of Bayesian statistics is Bayes’ Theorem:

\[ P(\theta \mid D) = \frac{P(D \mid \theta) \, P(\theta)}{P(D)} \]

Where:

  • ( \theta ) = parameters we want to estimate (e.g., S-curve shape parameters, nutrient effect coefficients).

  • ( D ) = observed data (e.g., GDGT fractions, Ring Index values).

  • ( P(\theta \mid D) ) = posterior — our updated beliefs about parameters given the data.

  • ( P(D \mid \theta) ) = likelihood — how well a given set of parameters explains the data.

  • ( P(\theta) ) = prior — what we believed about parameters before seeing the data.

  • ( P(D) ) = evidence — a normalization constant ensuring probabilities sum to 1.

3. Forward Model (Environmental Variables → Proxy Values)#

The forward model predicts scaled Ring Index (( RI_{\text{scaled}} )) from environmental predictors.

Example for a generalized logistic forward model:

\[ RI_{\text{scaled}}(T, X) = b + \frac{1 - b}{\left( 1 + e^{-k (T - T_{0})} \right)^{1/v}} + \sum_{j} \beta_j X_j \]

Where:

  • ( T ) = temperature.

  • ( X_j ) = additional predictors (e.g., GDGT-2/GDGT-3 ratio, nitrate concentration).

  • ( b ) = lower asymptote; the upper asymptote is fixed at 1; ( T_0 ) = inflection temperature, ( k ) = steepness, ( v ) = shape (curve asymmetry).

  • ( \beta_j ) = coefficients for optional predictors.

  • ( RI_{\text{scaled}} ) is normalized between 0 and 1 for comparability.

Forward Model Role:
Given true environmental conditions, the model produces a distribution of possible GDGT-based proxy values, including natural variability and measurement noise.

4. Bayesian Inference (Proxy Values → Environmental Variables)#

The inverse problem is: Given an observed proxy value (e.g., RI), what is the most likely temperature?

Using Bayes’ theorem:

\[ P(T \mid RI_{\text{obs}}) \propto P(RI_{\text{obs}} \mid T) \, P(T) \]

  • ( P(T) ) — prior temperature distribution (e.g., from regional climate models, modern analogs, or culture constraints).

  • ( P(RI_{\text{obs}} \mid T) ) — likelihood from the forward model.

  • The result ( P(T \mid RI_{\text{obs}}) ) is the posterior distribution of temperatures consistent with the observation.

4.1 Likelihood Function#

The likelihood assumes a statistical error model:

\[ P(RI_{\text{obs}} \mid T, \theta) = \mathcal{N} \left( RI_{\text{obs}} \,\middle|\, RI_{\text{pred}}(T, X; \theta), \sigma_{RI} \right) \]

  • ( RI_{\text{pred}} ) — forward model prediction.

  • ( \sigma_{RI} ) — residual standard deviation (accounts for unmodeled variability).

4.2 Priors#

  • From cultures — tightly constrain curve shape parameters (( T_0, k, v )) for known archaeal clades.

  • From core-tops — set broader priors for environmental effects (( \beta_j )).

Example prior for the temperature:

\[ T \sim \mathcal{N}(\mu_T, \sigma_T) \]

Where (\mu_T) and (\sigma_T) come from prior knowledge (e.g., modern climatology).

5. Putting It Together#

  1. Forward Step

    • Start with priors for all parameters.

    • Use the forward logistic model to simulate RI given environmental conditions.

    • Compare to observed RI values via the likelihood.

  2. Bayesian Updating

    • The posterior combines likelihood and priors.

    • We sample from the posterior (e.g., via MCMC in Stan/CmdStanPy).

  3. Inference Output

    • Posterior temperature distribution for each site/sample.

    • Full uncertainty range, not just a single “best estimate.”

6. Advantages for Paleoceanography#

  • Mechanistic: Based on GDGT biosynthesis behavior, not just empirical regression.

  • Flexible: Can include ecological and nutrient effects.

  • Transparent Uncertainty: Directly quantifies confidence intervals on temperature estimates.

  • Culture Integration: Uses lab data as informative priors, bridging modern process understanding with ancient climate reconstructions.

7. Key Equations Summary#

  1. Bayes’ Theorem $\( P(\theta \mid D) = \frac{P(D \mid \theta) \, P(\theta)}{P(D)} \)$

  2. Forward Logistic Model $\( RI_{\text{scaled}}(T, X) = b + \frac{1 - b}{\left( 1 + e^{-k (T - T_{0})} \right)^{1/v}} + \sum_{j} \beta_j X_j \)$

  3. Likelihood $\( P(RI_{\text{obs}} \mid T, \theta) = \mathcal{N} \left( RI_{\text{obs}} \,\middle|\, RI_{\text{pred}}, \sigma_{RI} \right) \)$

This workflow ensures that your TEXAS-PSM reconstructions combine the best available culture, modern, and paleo data in a statistically rigorous way, providing more reliable temperature estimates for past climates.