[Draft] Static Normals: A baseline predictive model

The visualization follows the style used in the previous post, which introduced the probabilistic NIR-2025. The visualization has been updated to show both a predictive model (the static normals model, conditioned on NIR-2025 data up to 2022) and the emissions that it should predict (the probabilistic NIR-2025's 2023 data, shown in terms of upper and lower credible interval and mean, thin solid and dotted black lines respectively).

This post also introduces predictions to the PlanZero site, in two ways. The first way is that visualizations of predictive probabilistic models show ranges of 95% credible emissions for future years. The second way is via the Predictions page, a new one on PlanZero. The predictions page provides an overview of kinds of prediction, and a ranking of models in terms of metrics that formalize how good they are at making those predictions. The previous post on the probabilistic NIR-2025 introduced a model evaluation metric for scoring predictions in terms of KL-divergence from the probabilistic interpretation of measured emissions. This post uses that metric to score the static normals model. The metric is used for both individual sector-gas combinations, as well as an overall weighted average across all sectors and gases. The static normals model scores an overall average divergence of 3.35. Future models might rank better by scoring closer to zero.

I believe this is the first step in an important research loop that can lead PlanZero to making the best possible predictions. The machine learning community has made great progress over recent decades by combining two incremental, mutually supportive research programs. The first one is a relatively slow accumulation of benchmark prediction challenges. The second one is a relatively faster exploration of the space of models, algorithms, and data to find ways of improving the state of the art in those challenges. At the same time, the more accurately PlanZero can predict emissions impacts, the more helpful PlanZero can be in evaluating the many strategies that have helpful but not enormous emissions impacts. This is why the PlanZero site is now structured to feature predictions and models that can reflect two mutually reinforcing research programs. I hope they will lead to as solid a foundation as possible for model-based planning of emission-reduction strategies. This rationale has been added to the About page.

Despite the simplicity of the static normals model, the predictions are qualitatively accurate for several sectors, and establish a numerical benchmark for quantitative scoring (i.e. divergences below about 1 are pretty good). The good fit for emissions sectors such as e.g. Enteric Fermentation, Municipal Solid Waste Landfills, Manure Management, and Railways suggests that little progress has been made in these areas, at least by 2023. Conversely, the poorness of fit in other sectors suggests that have been significant changes either for better or worse. We'll look at trends in a future post, that will introduce an autoregressive model as an alternative to the standard normals presented here. Before that post though, I'd like to insert two tidying-up posts. The Bovaer analysis should be adapted to operate on probabilistic models and the models page should be purged of "simulation-based models", which is a term I'd like to deprecate and remove from PlanZero. In a second tidy-up post, I'd like to to finally introduce the glossary page with confidence in my definition of what a "model" actually is in PlanZero.

Until next time,

- James Bergstra

Appendices

• Appendix 1: Probabilistic Modelling Notation
• Appendix 2: Model Description
• Appendix 3: Background in Bayesian Inference
• Appendix 4: Inference in the Standard Normals Model
• Appendix 5: Pre-NIR-2025-m04, a NIR Prediction Challenge
• Appendix 6: Update to PlanZero's Build System

Appendix 1: Probabilistic Modelling Notation

Some probability notation is conventional, in order to express probabilistic models. It builds on the introduction to probability in Appendix 1 of the Probabilistic NIR-2025 post. Probabilistic modelling is a huge topic. Basic concepts of probability are introduced across many years of primary and secondary schooling, and covered in greater detail at university levels. For more information and orientation, see e.g. wikipedia (Statistical Model, Graphical Models), many tutorials online that are specific to probabilistic programming software packages (e.g. Stan, PyMC, Pyro; PlanZero currently uses NumPyro) or a textbook such as Probabilistic Graphical Models (Koller and Friedman, 2009) for greater depth.

The following modelling notation used in this post to describe the Static Normals model and introduce Bayesian inference in Appendix 4: Inference in the Static Normals Model.

• $P(X=0)$ means there's a number we don't know, we're calling it an $X$, and this is the probability it is actually 0. $X$ is called an unknown variable in PlanZero, although many references refer to such a variable as a random variable.
• $P(X)$ is, intuitively, the table mapping all the values $X$ might take to their probabilities. The probabilities add up to 1 to represent the fact that $X$ must take one of them. The table might be infinitely large (e.g. a probability for each integer). Although it can be an abuse of notation, I'll also sometimes write $P(X)$ to refer to a function defining the density of probability mass when the possibilities for $X$ are a continuous domain of values, such as the real numbers.
• $P(0)$ is shorthand for $P(X=0)$ when it's clear from surrounding context we're talking about $X$
• $𝒩(\mu ,{\sigma }^2)$ is a Normal distribution, which is non-negative function that sums (integrates) to 1. It is parameterized by two numbers: $\mu$ the mean, and ${\sigma }^2$ the variance. It's conventional to write ${\sigma }^2$ as the argument to the normal distribution because $\sigma$ itself is qualitatively the "width" of the distribution and only positive-valued variances are valid. Normal distributions are widely used in probabilistic modelling. You can read more about them on wikipedia, and see its shape in the figure below. (Sometimes these are called Gaussians but I'm going to call them Normals to be consistent with the probabilistic programming library I'm using.)
• $\mathrm{L}\mathrm{o}\mathrm{g}\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}(\mu ,{\sigma }^2)$ is a log-normal distribution, a distribution over positive-valued real numbers with a given mean and relative uncertainty. There's an example below illustrating one possible shape of log-normal distribution, with $\mu =-1$ and ${\sigma }^2=\mathrm{.7}$. Note that unlike the regular normal, $\mu$ being negative doesn't make the distribution negative, it squeezes the distribution leftward toward 0. The ${\sigma }^2$ parameter doesn't exactly modulate the width, as it does for the normal distribution, but it reflects relative uncertainty, and it does affect both the shape and the mean of the distribution.
• $X~𝒩(0,1)$ means (1) that $X$ is an unknown variable in some probabilistic model, and (2) that its probability distribution (or density) $P(X)$ is the thing to the right of the "$~$" symbol. In this example, $X$ is distributed according to a normal distribution with mean 0 and variance 1.
• $P(A=0,B=1)$ features comma-separated terms. The equals-signs mean the terms are criteria. $P(A=0,B=1)$ means the probability that unknown variable $A$ is 0 and unknown variable $B$ is 1.
• $P(A,B)$ also features comma-separated terms, but the terms aren't predicates with equals-signs. This is the notation for the probability table (or function, as appropriate to the domains of $A$ and $B$) that maps every possible value of $A$ and every possible value of $B$ given that value of $A$, to the probability of them both taking those respective values. This sort of probability distribution is called a joint distribution because it is a distribution over multiple unknown variables.
• $P(A=0|B=1)$ uses the | symbol to mean the probability that $A=0$ if we make an additional assumption about $B$, namely that it is 1. $P(A|B=1)$ is called a conditional distribution because it defines a probability distribution for $A$ that is conditional on additional assumptions / criteria. A probabilistic model that includes $A$ and $B$ defines either explicitly or implicitly how they can be related by conditional distributions. Joint distributions such as $P(A,B)$ can always be written as products involving conditionals $P(A,B)=P(A|B)P(B)=P(B|A)P(A)$, and sometimes, such as when deriving techniques for Bayesian inference, such rewriting is very helpful.

Appendix 2: Model Description

PlanZero's "Static Normals" model of the 2025 National Inventory Report (NIR-2025, or just NIR in this post) is that for each ...

• IPCC sector (I'll call these "sectors")
• greenhouse gas (sometimes just "gas")
• province and territory (I'll call these "regions")
• and year (from 1990 to 2023)

Finally, and most radically, the model is called "static normals" because it assumes that these quantities of emissions don't change over time. Instead, it assumes that the different estimates of emissions from year to year in the NIR are the results of completely random fluctuations, totally unforseeable and outside of anyone's control, which will continue indefinitely into the future following the same Normal distribution of annual emissions as has been observed historically. This is a unrealistic assumption because in many sectors, the reported emissions are defined in the NIR (following IPCC Guidelines) as being proportional to e.g., the number of cars, or trucks, or heads of cattle, which are highly correlated from year to year, and which follow long-term trends. Yet, there's no rule against using unrealistic assumptions in models. The consequences of making unrealistic assumptions in models are uncertainty and/or inaccuracy. We'll use this model to establish a baseline for uncertainty and inaccuracy in prediction, which can serve as a benchmark for future models aiming for greater certainty and accuracy on the basis of e.g. greater realism.

Formally, the Static Normals model is the following set of distributions over unknown quantities:

Appendix 3: Background in Bayesian Inference

Bayesian inference in probabilistic models is about redistributing probability for unknown variables in order to balance two competing objectives: (1) to remain probable according to the a-priori distributions, and yet (2) to explain observations. The Static Normals model described in Appendix 2: Model Description includes a-priori probability distributions over $\mu$ and $\sigma$ that induce distributions over $X$ and $Y$, which are meant to estimate quantities in an NIR. Yet we can observe the NIR-2025, in terms of either raw data, or the distributions developed in PlanZero's NIR-2025 model (see Probabilistic NIR-2025 post for details). [Bayesian] Inference algorithms can answer the question: What are the conditional distributions over $\mu$ and $\sigma$ given certain observable values for $X$ and $Y$? This specific kind of conditional distribution, being a distribution conditioned on data, is called a posterior distribution. The unobserved variables in the model are called latent variables, and so a posterior distribution is a conditional distribution over the latent variables.

If only some of the observable data is used for conditioning, then the remaining data can be used to evaluate a model's predictions. For example, if we only condition on emissions data up to 2022, then variables related to year 2023 will be latent ones and inference will assign them posterior distributions. These posterior distributions can be interpreted as the model's predictions for 2023. Appendix 4: Pre-NIR-2026-m04, a NIR Prediction Challenge evaluates such predictions from the Static Normals model, using the method of inference discussed here in this Appendix.

Inference algorithms work on the basis of Bayes' Theorem. In standard probability notation, for any pair of unknown variables $A$ and $B$, Bayes' theorem looks like:

$$P(A|B)=\frac{P(A,B)}{P(B)}=\frac{P(B|A)P(A)}{P(B)}=\frac{P(B|A)P(A)}{{\int }_{a}P(B|A=a)P(A=a)}$$

Bayes' theorem shows how to relate a conditional distribution of the form $P(A|B)$ to conditional distributions the other way around: $P(B|A)$ and $P(A)$. Bayes' theorem is critical to inference in the Static Normals model because

• in order to make predictions, the Static Normals model was defined with the conditionals one way, namely $P(X,Y|\mu ,\sigma )$
• to observe NIR data, we need the opposite conditional distribution $P(\mu ,\sigma |X=\mathrm{N}\mathrm{I}{\mathrm{R}}_{(\mathrm{r}\mathrm{e}\mathrm{g}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{s})},Y=\mathrm{N}\mathrm{I}{\mathrm{R}}_{(\mathrm{n}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{s})})$

I don't know if it's possible to write down a formula for the posterior distribution $P(\mu ,\sigma |X=\mathrm{N}\mathrm{I}{\mathrm{R}}_{(\mathrm{r}\mathrm{e}\mathrm{g}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{s})},Y=\mathrm{N}\mathrm{I}{\mathrm{R}}_{(\mathrm{n}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{s})})$. For some models, there are closed forms for some posterior distributions, but it's not always possible. What's often practical, is the use of Monte Carlo (random) estimators to compute samples of points from the desired distribution, based only on the conditional distributions used to define the model. Even with a Ph.D. in machine learning, I still find these algorithms amazing (read up on e.g. MCMC algorithms or stochastic variational inference if you're interested in how these algorithms work). This intersection of statistics, machine learning, and computer science is still the subject of active research, but the core capability of sampling from posterior distributions is relatively mature and for the purposes of this post we'll take it as given.

Probabilistic programming systems enable someone such as myself to define a probabilistic model by simply implementing a sampling process (such as the one in the left column of the table in Appendix-2 that defined the model) in a programming language, and then request inference in that model with simple standardized function calls. The implementation of these function calls involves relatively sophisticated software techniques (e.g. tracing and runtime program transformations in order to run backward over the computations that defined the model) but the output is simple: samples from $P(\mu ,\sigma |X=\mathrm{N}\mathrm{I}{\mathrm{R}}_{(\mathrm{r}\mathrm{e}\mathrm{g}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l})},Y=\mathrm{N}\mathrm{I}{\mathrm{R}}_{(\mathrm{n}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l})})$. PlanZero uses the combination of Jax and NumPyro as a probabilistic programming system, and the NUTS MCMC algorithm in NumPyro for sampling from the posterior.

Appendix 4: Inference in the Static Normals Model

Running NumPyro's NUTS inference algorithm for the static model above, using the NIR-2025 emissions data up to 2022 for each sector and gas, and then re-sampling $Y$ according to the original model for visualization, given the posterior $P(\mu ,\sigma |X=\mathrm{N}\mathrm{I}{\mathrm{R}}_{(\mathrm{r}\mathrm{e}\mathrm{g}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l})},Y=\mathrm{N}\mathrm{I}{\mathrm{R}}_{(\mathrm{n}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l})})$, gives rise to the following figure, now available at the Static Normals model overview page. Inference for all models involved 250 warmup steps, followed by 500 samples with no thinning. Details of sampling can be found in nir_static_normals.py at time of writing.

flowchart TB

gh_build_and_test["Build and Test\n(Job)"]

gh_deploy_main["Deploy Main\n(Job)"]

gh_deploy_pr["Deploy Pull-Request\n(Job)"]

app_cache[("App Cache\n(Workflow Artifact)")]

inference_cache[("Model Inference Results\n(Cache Object)")]

inference_cache <-. "Per-model-version samples" .-> gh_build_and_test

gh_build_and_test --> gh_deploy_main
gh_build_and_test --> gh_deploy_pr

gh_build_and_test -. "Pre-rendered pages" .-> app_cache

app_cache .-> gh_deploy_main
app_cache .-> gh_deploy_pr