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Towards accurate extreme event likelihoods from diffusion model climate emulators

This paper demonstrates that probability density estimates from the diffusion model climate emulator cBottle can be used to calculate likelihoods of extreme events like tropical cyclones under guidance, enabling importance sampling to reduce estimation error and facilitating extreme event attribution experiments.

Original authors: Peter Manshausen, Noah Brenowitz, Julius Berner, Karthik Kashinath, Mike Pritchard

Published 2026-05-06

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Peter Manshausen, Noah Brenowitz, Julius Berner, Karthik Kashinath, Mike Pritchard

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 have a super-smart weather simulator, a digital "time machine" that can generate realistic pictures of the Earth's atmosphere. This simulator, called cBottle, is like a master chef who can cook up any weather scenario you want, as long as you give them the right ingredients (like ocean temperatures and the time of year).

Usually, if you ask this chef to make a "Tropical Cyclone" (a hurricane), they might say, "Sure, but I can only make one in a million tries." That's because hurricanes are rare. If you want to study them, you'd have to wait a very long time for the chef to accidentally cook one up.

This paper introduces a clever trick to speed things up and, more importantly, to know exactly how rare the hurricane really is.

The "Guided" Chef

The researchers found a way to "guide" the chef. Instead of just waiting for a hurricane to happen by chance, they can whisper, "Hey, put a hurricane right here over Miami."

When they do this, the chef instantly starts cooking up hurricanes. But here's the problem: If the chef is forced to make a hurricane, every single dish looks like a hurricane. You can't tell from the dishes alone how likely a hurricane was to happen naturally. It's like if a magician was forced to pull a rabbit out of a hat every time; you wouldn't know if pulling a rabbit is a normal trick or a miracle.

The "Odds Ratio" Calculator

The big breakthrough in this paper is a new math tool they built to answer the question: "How much more likely did we make this hurricane by whispering to the chef?"

Think of it like this:

The Natural Way: You ask the chef to make a random dish. You wait 1,000 times and get 1 hurricane. (1 in 1,000 chance).
The Guided Way: You whisper "Hurricane!" and get a hurricane every time.
The Calculation: The researchers developed a way to measure the "effort" the chef had to put in to obey the whisper. By measuring this effort, they can calculate a score (called an "odds ratio").

This score tells them: "Okay, because we whispered, this hurricane is 1,000 times more likely to appear than it would have been on its own."

By using this score, they can take their "forced" hurricanes and mathematically "discount" them. They can say, "We saw 1,000 hurricanes because we forced it, but after applying our discount score, we know that in the real world, this is actually a 1-in-1,000 event."

Why This Matters (The "Importance Sampling" Trick)

The paper calls this Importance Sampling. Imagine you are looking for a needle in a haystack.

The Old Way (Monte Carlo): You just start pulling out hay randomly. You might have to pull out a million pieces of hay before you find one needle.
The New Way (Guided + Discount): You use a magnet to pull out all the needles instantly. But because you used a magnet, you know you've pulled out too many needles. So, you use a calculator to figure out exactly how many "extra" needles the magnet gave you, and you subtract them.

The result? You find the needle much faster, and you still know exactly how rare it is. The paper shows that for the rarest, most extreme events, this method is actually more accurate than just waiting for them to happen naturally, because it reduces the "noise" in the data.

A Side Note: The "Student" Model

The researchers also tried using this math to look at real-world weather data (like a heatwave in Antarctica). They asked the simulator: "How likely is this specific heatwave?"

The Result: The simulator could tell them the heatwave was "unusual," but it couldn't perfectly predict how unusual it was based on temperature alone. It was like a student who knows the material but gets nervous during the test. The paper suggests this is because the simulator is still learning and needs to get better at understanding the "rules" of extreme weather.

The Catch: It's Expensive

There is one downside. Doing this math is computationally heavy.

The Analogy: It's like having a fast car (the simulator) but needing to drive it in slow motion to measure the engine's vibration perfectly.
The Cost: To get one "guided" hurricane with this perfect math score, it currently takes about 33 times longer than just letting the simulator run normally.
The Future: The authors hope that in the future, they can make this math faster (like "distilling" the knowledge), so it becomes a practical tool for climate scientists.

Summary

In short, this paper shows that we can use AI to force a climate model to create extreme weather events (like hurricanes) so we can study them easily. Then, using a special math trick, we can undo the forcing to calculate exactly how rare those events are in the real world. It's a way to study the "impossible" without losing track of the "probable."

Technical Summary: Towards Accurate Extreme Event Likelihoods from Diffusion Model Climate Emulators

Problem Statement
Machine learning (ML) climate emulators, particularly diffusion models like Climate in a Bottle (cBottle), offer cost-efficient experimentation for scenario planning and adaptation. A key capability of these models is "guidance," where they can be steered to generate specific states of interest, such as Tropical Cyclones (TCs) in user-defined locations. However, a critical limitation exists: while guided models can oversample rare extreme events, they do not inherently inform users about the likelihood of these generated samples under the true, unguided data distribution. Without this probabilistic information, it is difficult to distinguish between plausible extremes and artifacts of the guidance process, limiting the utility of these models for rigorous extreme event attribution and risk assessment.

Methodology
The authors propose a framework to recover accurate probability density estimates for guided samples using the cBottle diffusion model. The methodology relies on the theoretical foundation of Denoising Diffusion Probabilistic Models (DDPMs) and the Probability Flow (PF) Ordinary Differential Equation (ODE).

Guided Sampling: The cBottle model, a diffusion-based climate emulator at 5km resolution, is conditioned on sea surface temperatures (SSTs) and solar position. To generate TCs, a classifier guidance term is added to the PF-ODE. The authors refine the guidance implementation by:
- Using a constant guidance scale ( $\gamma$ ) rather than the adaptive scaling used in previous work.
- Applying guidance only within a specific "denoising window" (noise levels $\sigma \in [15, 20]$ ) where the TC signal is generated, rather than across the entire trajectory. This avoids implausible divergence values and reduces computational cost.
Odds Ratio Calculation: To quantify how much the guidance alters the likelihood of a sample, the authors calculate the odds ratio $o(x) = p_{\text{unguided}}(x) / p_{\text{guided}}(x)$ .
- Using the PF-ODE formulation, they compute the log-probability density of a sample by integrating the divergence of the velocity field from the sample back to the Gaussian prior.
- The log odds ratio is decomposed into three terms: the difference in latent probabilities ( $\Delta_{\text{latent}}$ ), the difference in denoiser score divergences ( $\Delta_{\text{denoiser}}$ ), and the divergence of the guidance term ( $\Delta_{\text{guidance}}$ ).
- These divergences are estimated using the Skilling-Hutchinson trace estimator with automatic differentiation.
Importance Sampling: The calculated odds ratios are used to perform importance sampling. Instead of drawing $K$ samples from the unguided distribution (which rarely yields TCs), the authors draw samples from the guided distribution and reweight them by the odds ratio. This allows for the estimation of the true probability of extreme events with reduced standard error compared to simple Monte Carlo sampling.
ERA5 Likelihood Analysis: The authors also apply the unguided model to calculate the likelihood of real-world ERA5 reanalysis data (specifically Antarctic temperature anomalies) to test the model's response to SST perturbations and extreme events.

Key Results

Guidance Window Optimization: The study demonstrates that guidance is only effective and numerically stable within a narrow noise interval ( $\sigma \approx 15-20$ ). Applying guidance across the full trajectory leads to negative divergence values and implausible samples. Restricting guidance to this window yields stable TC generation with positive divergence.
Variance in Odds Ratios: The authors observe that the variance of the calculated odds ratios increases significantly with stronger guidance strength. While strong guidance produces more TCs, the resulting weights become highly variable. Consequently, the most robust importance sampling estimates are achieved with weaker guidance scales, requiring repeated sampling to reduce variance.
Accurate Likelihood Recovery: Despite the variance, the importance sampling estimates successfully recover the TC frequency of the unguided control model. For extreme thresholds (e.g., 99.999th percentile), the importance sampling estimate achieves lower uncertainty than a standard Monte Carlo estimate from the unguided model, effectively trading sampling uncertainty for weight uncertainty.
Comparison with ERA5: When comparing cBottle's TC climatology to ERA5 historical records, the spatial patterns match well, though cBottle underestimates the absolute frequency and confidence of TCs. The authors note that this bias is inherent to the model's current training and architecture, not the guidance or sampling method.
ERA5 Sample Probabilities: For the March 2022 Antarctic heatwave, the model assigns typical log-probability values for that time of year, showing a pronounced seasonal cycle (higher probability in summer, lower in winter) rather than a direct correlation with the magnitude of the temperature anomaly. The model does correctly assign higher probabilities to heat events under +2K SST perturbations compared to -2K, but the log-probability does not effectively serve as a direct proxy for the extremity of the event itself.

Significance and Claims
The paper presents "early but encouraging results" demonstrating that diffusion model climate emulators can provide not just synthetic samples of extreme events, but also accurate probabilistic estimates of their likelihoods.

Novelty: To the authors' knowledge, this is the first application of calculating odds ratios as the ratio of sample probabilities under guided versus unguided diffusion models to enable importance sampling for climate extremes.
Utility: The method allows researchers to oversample rare events (like TCs) for impact studies while maintaining statistical rigor regarding their true occurrence likelihood.
Limitations and Future Work: The authors are modest about the current computational cost, noting a 33x slowdown for a single guided sample with odds ratio calculation compared to unguided sampling. They suggest that future improvements, such as model distillation or classifier-free guidance, could make this approach more cost-effective. They also emphasize that the accuracy of the likelihoods is fundamentally limited by the fidelity of the underlying cBottle model, particularly its representation of TC strengths.

The work aims to spur further research into extracting probabilistic information from diffusion models, positioning them as reliable tools for climate risk assessment and scenario planning.