🤖 machine learning

Never mind the metrics -- what about the uncertainty? Visualising confusion matrix metric distributions

This paper argues that the substantial uncertainty inherent in empirical classifier performance metrics often eclipses observed differences in model accuracy, advocating for a balanced perspective through novel visualizations of confusion matrix distributions and posterior predictive probabilities in ROC space.

Original authors: David Lovell, Dimity Miller, Jaiden Capra, Andrew Bradley

Published 2026-06-29

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: David Lovell, Dimity Miller, Jaiden Capra, Andrew Bradley

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

The Big Idea: Don’t Trust the Scorecard Blindly

Imagine you are judging a shooting competition. One shooter hits the bullseye 9 times out of 10 shots. Another hits it 8 times out of 10. On paper, the first shooter looks better.

But what if the first shooter only took 10 shots total, while the second shooter took 1,000 shots? The first shooter’s score is shaky; they might have just gotten lucky. The second shooter’s score is solid; we know they are truly skilled.

This paper argues that in the world of Artificial Intelligence (AI) and machine learning, we are too obsessed with the "score" (the performance metric) and not paying enough attention to the "luck factor" (the uncertainty). The authors want us to stop arguing about which score is "best" and start understanding how much we can actually trust that score based on the amount of data we have.

The Problem: The "Single Number" Trap

When an AI tries to sort things into two categories (like "spam" vs. "not spam"), we use a tool called a Confusion Matrix. It’s a simple table that counts:

How many it got right.
How many it got wrong.

Scientists then crush this whole table down into a single number, like Accuracy or MCC (Matthews Correlation Coefficient). The culture in AI is to chase the highest number. If Model A has a score of 0.95 and Model B has 0.94, everyone declares Model A the winner.

The authors say this is dangerous. Why? Because that single number hides the fact that the data is discrete (chunky) and finite (limited). You can’t have half a correct prediction. You have whole numbers. And because you only have a limited number of examples, your score is just an estimate, not a fact.

The Solution: Visualizing the "Fuzziness"

The paper introduces a way to see the "fuzziness" or uncertainty behind those scores. They use two main tricks:

1. The 3D Tetrahedron (The Confusion Simplex)

Imagine all possible outcomes for an AI test as points inside a 3D pyramid (a tetrahedron).

Each corner of the pyramid represents a perfect score for one specific type of outcome (e.g., all correct positives).
Every possible test result is a dot inside this pyramid.

The authors show that instead of looking at flat 2D charts, we can look at this 3D shape. This helps us see how the "shape" of the data changes when one category is rare (class imbalance). It’s like looking at a map from above versus looking at a 3D terrain model; the 3D view shows you the hills and valleys (the nuances) that the flat map hides.

2. The "Cloud of Possibility" (Uncertainty Visualization)

This is the core contribution. The authors use math (specifically Bayesian statistics) to answer this question: "If we ran this test again with new data, what scores might we get?"

They don’t just give you one score. They give you a distribution—a cloud of possible scores.

Small Data: If you only tested the AI on 20 items, the "cloud" is wide and fuzzy. The AI might score 0.9 today, but 0.7 tomorrow. The uncertainty is high.
Large Data: If you test it on 10,000 items, the "cloud" shrinks to a tiny, sharp point. The score is stable.

They created interactive tools (like digital graphs you can play with) to show this. They show that for small datasets, the "cloud" of possible scores for two different AIs might overlap completely. Even if one AI has a slightly higher average score, the uncertainty is so big that you can’t really say one is better than the other.

The Key Takeaway: More Data, Less Drama

The paper makes three main points:

Metrics are Smooth, Data is Chunky: Performance scores (like MCC) look like smooth curves on a graph, but the actual data is made of whole numbers (you can’t have 3.5 correct answers). This mismatch creates "jumps" in the data that standard charts hide.
Uncertainty is Real: When you have small amounts of data, or when one category is very rare (like detecting a rare disease), your performance score is highly uncertain. A score of 0.8 might actually be anywhere between 0.5 and 0.9.
Stop Arguing, Start Collecting: The authors argue that scientists spend too much time debating which metric (MCC vs. F1 vs. Accuracy) is "best." They say this is a distraction. The real solution to uncertainty isn’t picking a better metric; it’s getting more data.

A Creative Analogy: The Weather Forecast

Think of an AI performance metric like a weather forecast.

The Metric: "There is an 80% chance of rain."
The Uncertainty: If the meteorologist only looked at 3 days of weather history, that 80% is a guess. It could easily be 50% or 100%. The "cloud" of uncertainty is wide.
The Data: If the meteorologist looked at 100 years of weather history, that 80% is very precise. The "cloud" is tight.

The paper says: Don’t argue about whether "80%" is a better number than "79%." Ask instead: "How many years of weather data did you use to get that number?" If the data is thin, the number is unreliable, no matter how impressive it looks.

Conclusion

The authors want researchers to be humble. When you see an AI model with a great score, ask: "How uncertain is this score?" If the dataset is small, the score is likely noisy. The paper provides tools to visualize this noise, hoping that scientists will stop treating single numbers as absolute truths and start treating them as estimates that need more data to become reliable.

Technical Summary: Visualising Confusion Matrix Metric Distributions

Problem Statement
The paper addresses the prevailing culture in algorithmic modeling that prioritizes and rewards models based on single-number performance metrics (e.g., Accuracy, F1, MCC). The authors argue that this focus risks a narrow perspective, ignoring the substantial uncertainty inherent in empirical performance estimates derived from finite datasets. Current literature often engages in debates over which metric is "best," but these discussions frequently overlook the fact that performance metrics are estimates subject to variance. The core problem identified is that uncertainty in confusion matrix counts—particularly when data is limited or classes are imbalanced—can eclipse the empirical differences between classifiers. The authors contend that practitioners need a better understanding of this uncertainty to temper performance claims and avoid over-interpreting minor metric differences.

Methodology
The authors employ a geometric and probabilistic approach to visualize and quantify uncertainty in binary classification performance.

Geometric Representation (The Confusion Simplex):
- The paper critiques existing projections (such as the confusion tetrahedron by Chicco et al.) for losing information about absolute counts and treating the space as continuous.
- Instead, the authors propose an isometric projection of binary confusion matrices into a three-dimensional compositional lattice, termed the "confusion simplex." This projection preserves the discrete nature of the counts and the information regarding the total number of examples ( $N$ ).
- By slicing this 3D simplex, the authors derive 2D rectangular lattices that correspond to Receiver Operating Characteristic (ROC) space. This allows for the visualization of performance metric contours within and beyond the standard ROC space $[0,1] \times [0,1]$ .
Algebraic Derivation of Contours:
- The authors derive exact algebraic expressions for the contours (isolines) of various performance metrics, including prevalence-dependent metrics (e.g., Accuracy, F1, MCC) and prevalence-independent metrics (e.g., Balanced Accuracy, Bookmaker Informedness).
- These equations allow for the precise visualization of how metric values change with True Positive Rate (TPR) and True Negative Rate (TNR), revealing geometric properties such as the elliptical contours of MCC.
Probabilistic Modeling of Uncertainty:
- To model uncertainty, the authors utilize Bayesian statistics. They treat the True Positive Rate and True Negative Rate as random variables.
- They compare two models:
  - Binomial Model: Assumes the observed rates are the true parameters.
  - Beta-Binomial Model: Incorporates prior uncertainty (using uniform Beta(1,1) priors) to generate posterior predictive distributions. This model is more conservative and reflects the additional uncertainty inherent in estimating rates from finite samples.
- The authors calculate the exact discrete probability mass functions (PMFs) for the posterior predictive distributions of confusion matrices, rather than relying on simulation-based approximations (like MCMC).
Visualization of Metric Distributions:
- The PMFs of the confusion matrices are mapped onto the performance metric contours. The distribution of a performance metric is derived by summing the probability masses of all confusion matrices that yield the same metric value.
- Interactive visualizations (via Desmos) and animations are provided to show how these distributions change with sample size ( $N$ ) and class balance ( $p$ vs. $n$ ).

Key Contributions

The Confusion Simplex: A novel isometric 3D projection that preserves count information and highlights the discrete structure of confusion matrices.
Exact Contour Equations: Algebraic derivations for the contours of numerous performance metrics, enabling precise visualization of their geometry in ROC space.
Discrete PMF Visualization: A method to calculate and visualize the exact discrete posterior predictive PMFs of performance metrics (MCC, BA, F1) under binomial and beta-binomial uncertainty models.
Interactive Tools: The provision of open-source interactive visualizations and animations that allow practitioners to explore the impact of sample size and class imbalance on metric uncertainty.

Results

Discreteness and Jumps: The visualizations reveal that performance metric distributions are not smooth continuous curves but are composed of discrete probability masses. The shape of these distributions depends on the alignment of the discrete ROC lattice points with the continuous metric contours. For example, adding a single negative example can significantly alter the PMF shape for MCC and Balanced Accuracy by changing which lattice points lie on specific contours.
Magnitude of Uncertainty: The paper demonstrates that uncertainty in performance metrics is substantial, especially with small sample sizes. The spread of the posterior predictive PMF can be wide, indicating that the observed metric value is just one of many plausible outcomes.
Impact of Data Size: As the total number of examples ( $N$ ) increases, the standard deviation of the TPR and TNR PMFs decreases proportionally to $1/\sqrt{N}$ , leading to more precise metric estimates.
Class Imbalance: The geometry of metric contours (e.g., the eccentricity of MCC ellipses) changes with prevalence. However, the authors emphasize that the fundamental issue is not the imbalance itself, but the absolute number of examples in the minority class, which drives uncertainty.

Significance and Claims
The authors claim that this work provides a more balanced perspective on classifier evaluation by shifting focus from comparing single-point metric estimates to understanding the distributions of those estimates. They argue that:

Uncertainty is Primary: The primary cause of uncertainty in performance metrics is the finite nature of evaluation data, not the choice of metric. No metric can reduce this uncertainty; only more representative data can.
Tempering Claims: Performance claims should be tempered by an awareness of the underlying uncertainty. Differences in metric values between classifiers may be statistically insignificant given the width of their predictive distributions.
Beyond "Best" Metrics: The debate over which metric is "best" is secondary to the need for rigorous uncertainty quantification. Practitioners should use these visualizations to assess the reliability of their evaluations, particularly in scenarios with rare classes or limited data.
Educational Value: The visualizations serve as a tool to help practitioners intuitively grasp the relationship between confusion matrix counts, ROC space geometry, and metric uncertainty, encouraging more careful interpretation of benchmark results.