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Bayesian Functional Emulation of CO2 Emissions on Future Climate Change Scenarios

This paper proposes a fully Bayesian functional regression emulator with a mixed effects hierarchical model and autoregressive error covariance to enable continuous time evaluation of climate-economy integrated assessment model ensembles for CO2 emissions and future climate change scenarios.

Original authors: Luca Aiello, Matteo Fontana, Alessandra Guglielmi

Published 2026-06-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Luca Aiello, Matteo Fontana, Alessandra Guglielmi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you want to know what the weather will look like in 50 years, but instead of looking at the sky, you have to ask five different, extremely complex supercomputers for their opinion. These computers are called Integrated Assessment Models (IAMs). They are like five different chefs trying to predict the taste of a soup that hasn’t been cooked yet. Each chef uses different recipes (assumptions about the economy, population, and technology) and has their own unique style.

The problem is that these "chefs" are incredibly slow and expensive to run. They only give you their answer every ten years (like a snapshot at 2020, 2030, 2040, etc.), and they disagree with each other. If you want to know what happens in 2025, you’re stuck guessing.

This paper proposes a clever statistical shortcut. Instead of running the expensive supercomputers over and over, the authors built a "Statistical Emulator." Think of this emulator as a smart, fast, and cheap "stand-in" or a digital twin that learns the habits of the five original supercomputers.

Here is how they did it, explained with everyday analogies:

1. Turning Snapshots into a Smooth Movie

The original supercomputers only give data points every decade. It’s like watching a movie where you only see one frame every ten seconds. You can’t really see the action in between.
The authors used a mathematical technique called Functional Data Analysis. Imagine taking those few blurry snapshots and drawing a smooth, continuous line through them. Now, instead of just knowing the CO2 levels in 2030 and 2040, the emulator can tell you what the level probably was in 2035, 2036, or any specific day in between. They call this "temporal downscaling"—filling in the blanks to make the data smooth and continuous.

2. The "Bayesian" Way of Thinking

The authors used a method called Bayesian Statistics. In everyday terms, this is like updating your beliefs as you get new information.

Before seeing the data: You have a "prior" guess (a hunch) about how CO2 emissions might behave.
After seeing the data: You look at what the five supercomputers actually said and update your hunch.
The Result: You don’t get just one single number for the future. You get a range of possibilities with probabilities attached. It’s not just saying "CO2 will be 50 gigatons"; it’s saying "There is a 95% chance CO2 will be between 48 and 52 gigatons." This gives policymakers a much clearer picture of the uncertainty involved.

3. Finding the Key Drivers

The emulator helped the authors figure out which factors actually matter most when predicting CO2 emissions. They looked at five main ingredients:

Population (POP)
GDP per capita (GDPPC) – How rich the world is.
Energy Intensity (END) – How efficiently we use energy.
Fossil Fuel Availability (FF) – How much coal/oil/gas is available.
Low-Carbon Tech (LC) – How much we develop green technology.

What they found:

The Big Players: The two most important factors were GDP per capita and Energy Intensity.
- Surprisingly, as the world gets richer (higher GDP), the model suggested a negative effect on emissions in their specific mathematical setup (likely due to how the scenarios were structured, where richer societies in certain scenarios might adopt cleaner tech faster or have different consumption patterns, though the paper notes this needs careful interpretation).
- Improving energy efficiency (using less energy to do the same work) was a huge driver.
The Latecomers: Population and Fossil Fuel availability became significant later in the century (after 2050–2070).
The Surprise: The development of Low-Carbon Technology (LC) didn’t seem to have a statistically significant impact on the emissions in this specific model. The authors suggest this might mean that simply inventing green tech isn't enough; we also need to change how much we consume and our lifestyles.

4. Why This Matters

Currently, scientists look at the range of answers from the five different supercomputers and just say, "Well, the answer is somewhere between the highest and lowest prediction." This is a rough, "empirical" guess.

This paper’s emulator is better because:

It’s fast: You can ask it questions instantly without waiting for supercomputers to run.
It’s smooth: It gives you answers for any year, not just decades.
It’s probabilistic: It tells you how sure it is about its answers, giving a true measure of uncertainty rather than just a raw range.

In Summary

The authors created a smart, fast, statistical "stand-in" for five slow, expensive climate-economy supercomputers. By using Bayesian math, they turned choppy, decade-by-decade data into a smooth, continuous story of future CO2 emissions. They found that economic wealth and energy efficiency are the biggest drivers of future emissions, while green technology alone didn't show up as a major factor in their model. This tool helps policymakers see not just what might happen, but how uncertain those predictions are, allowing for better, more informed decisions.

Technical Summary: Bayesian Functional Emulation of CO2 Emissions on Future Climate Change Scenarios

Problem Statement
The paper addresses the challenge of quantifying uncertainty in Integrated Assessment Models (IAMs), which are deterministic computer simulators used to project future climate change impacts, specifically carbon dioxide ( $CO_2$ ) emissions. IAMs are computationally expensive and complex, often producing outputs only at discrete time intervals (e.g., every ten years). Furthermore, different IAMs yield varying projections for the same input scenarios due to differences in model implementation and assumptions. The authors aim to develop a statistical emulator that can:

Capture the variability induced by different IAMs and input parameters.
Provide a continuous functional representation of the discretized output, allowing for evaluation at any time point (temporal downscaling).
Quantify uncertainty probabilistically through a Bayesian framework, rather than relying on empirical ranges of deterministic outputs.

Methodology
The authors propose a Bayesian Functional Regression framework, specifically Function-on-Scalar Regression (FOSR), to emulate the ensemble of IAM outputs.

Data Structure: The data consists of $CO_2$ emission profiles (log-transformed) from 5 different IAMs ( $J=5$ ) across 23 combinations of Shared Socio-economic Pathways (SSPs) ( $I=23$ ). Each profile is discretized at 8 time points ( $D=8$ , representing decades from 2020 to 2090). The inputs are scalar representations of SSP variables (Population, GDP per capita, Energy intensity, Fossil fuel availability, Low-carbon technology).
Model Specification: The response $y_{ij}(t)$ for IAM $j$ and SSP combination $i$ at time $t$ is modeled as:
$y_{ij}(t) = w_i^T \beta(t) + c_i(t) + \epsilon_{ij}(t)$
where $w_i$ is the vector of SSP inputs, $\beta(t)$ represents the fixed regression coefficient functions, $c_i(t)$ are random effect coefficient functions specific to each SSP combination, and $\epsilon_{ij}(t)$ is the error term.
Functional Basis Expansion: To handle the functional nature of the response, $\beta(t)$ and $c_i(t)$ are expanded using a truncated B-spline basis with $K=8$ components. This reduces the infinite-dimensional problem to a finite-dimensional one involving basis scores.
Hierarchical Bayesian Model: A mixed-effects hierarchical model is employed. The random effects $c_i(t)$ allow for "borrowing of information" across groups, improving estimates for smaller groups. The model is hierarchically centered to improve MCMC sampling efficiency.
Covariance Structure: The error term $\epsilon_{ij}$ is assumed to follow a multivariate normal distribution with an autoregressive AR(1) covariance structure $\Sigma$ . This choice was made for computational efficiency and to facilitate temporal downscaling, as opposed to a full covariance matrix.
Inference: Posterior inference is performed using Stan via MCMC simulations. Priors are specified for all parameters, including hyperparameters for the variance components and the autoregressive correlation parameter.

Key Contributions
The authors claim five-fold original contributions:

Proposal of a Bayesian model for inference on $CO_2$ simulated emissions, a novel approach for this type of simulator data.
Specific choice of the Bayesian model structure (likelihood and priors).
Probabilistic interpretation of results, including a measure of how much each SSP contributes to the output within the IAMs.
Modeling the variability of single SSP combinations induced by different IAMs, providing credible intervals for new simulator outputs.
Providing a continuous framework for discretized simulation outputs, enabling emission evaluation at any time point, not just the discrete decades provided by the IAMs.

Results

Regression Coefficients: The posterior analysis identifies Gross Domestic Product per capita (GDPPC) and Energy intensity improvements (END) as the principal driving fixed effects. GDPPC has a negative effect (consistent with richer populations potentially consuming more, though the sign interpretation in the text notes a negative coefficient which the authors explain via scenario assumptions), while END has a positive effect. Fossil fuel availability (FF) and Population (POP) become significant later in the century (after 2050 and 2070, respectively). Low-carbon technology development (LC) is found to be statistically insignificant.
Emission Curves: The model successfully estimates the posterior mean of the random effects $c_i(t)$ , providing smooth emission curves with 95% credible intervals. These intervals capture the uncertainty inherent in the multi-IAM framework, offering a probabilistic interpretation superior to simple min-max ranges of deterministic outputs.
Temporal Kriging: The authors demonstrate temporal downscaling by predicting emissions at mid-decade points (e.g., 2025, 2035). The observed data points lie within the 95% credibility intervals of the estimated mean curves, validating the emulator's predictive capability.
Performance: The posterior predictive Mean Squared Error (MSE) of the Bayesian model (0.016) is comparable to that of a frequentist functional data analysis approach (0.014). The authors argue that despite the slight difference in MSE, the Bayesian approach offers superior interpretability and uncertainty quantification.

Significance
The paper claims significance in providing a unified Bayesian framework for emulating multiple IAMs simultaneously. This allows policymakers and climate scientists to:

Obtain uncertainty intervals with a clear probabilistic meaning.
Interpret results immediately through the posterior distributions of parameters.
Evaluate emissions at any time point, reducing the computational burden of running expensive simulators at high temporal frequencies.
Understand the relative contributions of different socio-economic pathways and IAM structural uncertainties to $CO_2$ projections.

The authors emphasize that this work moves beyond the standard empirical evaluation of IAM outputs by providing a stochastic representation that captures both parametric and model uncertainty in a coherent probabilistic framework.

1. Turning Snapshots into a Smooth Movie

2. The "Bayesian" Way of Thinking

3. Finding the Key Drivers

4. Why This Matters

In Summary

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