The Octagon Is a One-Trial Experiment
Jon Jones was, by almost any measure, the most complete mixed martial artist ever assembled. In 2023, he returned from a three-year layoff to fight Ciryl Gane for the heavyweight championship, and he won by submission in the first round of a fight that was not particularly close.
Now, let’s imagine that we had never seen Jones fight, that we watched exactly one round of that contest, and that someone then asked us: what is the probability Jones wins this match?
For a frequentist, this is a problem. Frequency-based probability is formally defined as a limiting ratio (the number of times an event occurs divided by the number of opportunities, as the number of opportunities goes to infinity). A single fight is one trial, and we cannot take a limit over one data point. Thus, strictly speaking, the frequentist cannot assign a probability to this specific contest.
I don’t think this is a minor inconvenience. It is one of the oldest and most discussed objections in the philosophy of probability, usually called the single-case problem, and it is treated at length by Hájek (2009) and Gillies (2000).
However, most people watching that fight thought Jones was a heavy favorite, and they were right. I’d argue that they were right in a way that a frequentist framework, properly applied, can actually explain, although the explanation looks a bit different from what the objection implies.
The standard critique goes something like this. Frequentism requires repeatable experiments, and a unique event (this fight, on this night, with this version of these two people) can’t be repeated. Therefore, frequentism is useless for one-off events. Bayesians, by contrast, treat probability as a degree of belief updated on evidence, which works fine for singular events.
The critique is fair as far as it goes. However, it seems to conflate two things, which are the probability of a fight type and the probability of this specific fight.
When a model estimates that Jones wins a matchup like his bout with Gane with, say, a 70% probability, it makes no claim about a unique and unrepeatable event. The claim is about a class of contests (fights between a wrestler of Jones’s profile and a striker of Gane’s profile, under roughly these conditions). In other words, the probability is a statement about the reference class, and the individual fight only inherits it.
This is how von Mises originally framed it, where probability is always about collectives and never about single instances. So what we need is the right collective, and (as we’ll see below) choosing that collective is the hard part.
The empirical floor: what the betting market knows
Betting markets in MMA aggregate the predictions of many informed people, and their track record gives us a useful empirical anchor.
The numbers in this section come from market data and were not produced by controlled experiments. They describe the aggregate accuracy of odds-setters and bettors, and they say nothing direct about the underlying probability of any single fight.
Across all UFC fights, favorites win somewhere in the range of 60 to 65 percent of the time, which means that underdogs pull off upsets in more than a third of all fights. That 35-to-40 percent upset rate is notably higher than what we see in most major team sports.
Heavy favorites do better. UFC odds priced between -400 and -900 (implying an 80-to-90 percent win probability) have returned an 88 to 93 percent win rate since 2013. At the other end, fights where the odds land between +100 and -122 (roughly a coin flip) have resolved in favor of the nominal favorite only 51 percent of the time.
Thus, the market does seem to predict fight outcomes with meaningful accuracy when there is a real skill gap between the fighters. When the gap is small, its predictions fail, or at least flatten out.
Skill is real; one fight is still noisy
Here’s the central claim, stated as plainly as I can: a fighter being more skilled than their opponent does not mean they win every fight. It means that their probability of winning a single fight is greater than 0.5, and perhaps substantially so. However, a 70% win probability is also a 30% loss probability.
This may sound trivial, but I don’t think it is. It is, in miniature, the same confusion we see when applied researchers treat a statistically meaningful estimate as proof that the effect always occurs. The estimate describes a distribution, and a single draw from that distribution can land almost anywhere within it.
When Georges St-Pierre was 170 pounds of organized violence in his prime, his aggregate probability of winning any given welterweight fight was very high. However, any single fight carried:
- Opponent stylistic advantages that a model underweights
- Chance of injury at any moment
- Variance in execution (did the game plan hold? did the cuts go as expected?)
- Pure stochastic noise (scrambles in grappling are notoriously non-deterministic)
A single punch, a well-timed submission, or a referee stoppage can override any statistical edge a fighter might carry on paper, and a scramble on the mat can convert a dominant position into a finished fight in one second.
If we ran that version of GSP against a solid but beatable opponent ten times, we’d expect him to win eight or nine of them. In one fight, we’d still expect him to win, although “expect” here is a probabilistic statement and comes with no guarantee.
The reference-class problem is the real fight
Choosing the right reference class is where frequentist analysis actually gets difficult in MMA, and it is probably where most naive applications go wrong.
Holmes, McHale, and Żychaluk (2023) proposed a Markov chain simulation approach that explicitly estimates fighters’ skills in key aspects of the sport (striking, grappling, clinch work, etc.) and then simulates the fight, instead of simply predicting a binary outcome. Why does this matter? Because the outcome of a fight is a poor fit for a single draw from a bivariate distribution. It is the aggregate result of hundreds of micro-decisions, each of which has its own stochastic component.
When we model it that way (as they did, comparing favorably to bookmaker odds), we are treating the fight as a sequence of probabilistic events from a well-defined class. Here, the frequentist framework is doing real work, and it is simply being applied at the level of techniques and positions instead of whole fights.
I think this is a defensible and scientifically useful construction. It is also the same move that statisticians make in clinical medicine, where we cannot run a randomized trial on a specific patient, but we can estimate their probability of response from a well-defined reference population and then use that estimate to guide the decision.
What this means for how you talk about fight outcomes
The mistake that most analysts (and most fans) seem to make is a binary one, where the favorite “should” win, the underdog “shouldn’t,” and an upset is a surprise that requires an explanation.
A better way to look at it is that a fight between a 70% favorite and a 30% underdog has no predetermined outcome and is a draw from a distribution with that shape. Three out of ten times, the underdog wins, and when that happens we don’t need a special explanation for it, since it is what we should expect the distribution to produce.
An upset does warrant an explanation if upsets happen at a rate higher than the odds imply (i.e., if the market is miscalibrated). The Fightful analysis of heavy favorites suggests that the market is reasonably calibrated at the extremes and noisier in the middle.
The key methodological point is that a single fight is a sample of size one from a fighter’s underlying skill distribution. Skilled fighters have higher-mean distributions with less variance, while fighters with fight-ending power but defensive liabilities have wide distributions. Thus, the outcome of one draw does tell us something, but probably not very much.
This is why it is statistically incoherent to say, after a single upset, that the underdog was “always better.” It is equally incoherent to say, after a dominant win, that the favorite “always wins this,” since in both cases we would be reading a great deal into one draw.
The Bayesian objection, and why it doesn’t fully resolve things
Bayesians handle this better in some respects. They assign a prior belief about the outcome, update on the evidence, and end up with a posterior. For singular events, this seems to be the most coherent framework, since a posterior probability represents a degree of belief about this specific contest and does not need a reference class to be meaningful.
However, the Bayesian approach faces its own problem in combat sports, which is that we need a prior, and priors in MMA are hard to specify. Two fighters may never have faced similar opponents, may have competed in different promotions, and may have trained specifically for this contest in ways the market cannot observe.
In practice, I think the most useful approach is an instrumental one, with less concern for doctrine. We build the best reference class we can using comparable fights and fighters, fit a model, and interpret the output as a probability statement about the type of contest this represents, while acknowledging that any single fight is an unreliable single observation of the underlying process.
That is what frequentism looks like in practice. It does not claim that probabilities can’t apply to single fights, and it only demands that we ground the probability in something empirically defined.
The upshot
MMA really is less predictable than most sports. The upset rate is substantially higher than in major team sports, and the mechanisms seem clear enough, since it is a sport defined by rapid phase transitions (standup, clinch, ground), each of which carries its own variance.
I wouldn’t treat that unpredictability as a flaw in the analysis, because it is a property of the process itself. A skilled fighter’s edge is real and measurable in the aggregate (the betting market shows this plainly at heavy price differentials). However, it is not deterministic in a single contest.
The fighter who “should” win is the one with the higher probability under the best available model. Whether they win this time is a separate question, and anyone who collapses the two has made the same mistake that researchers make when they treat a significant p-value as proof that the effect always occurs.
As a rule, then, I’d treat one fight as one trial, and I’d look to the distribution (and the reference class behind it) before reading much into any single result.
See also: Semantic and Cognitive Tools to Aid Statistical Science, which makes the case for replacing “significant” with “compatible” and describing what the data are more and less consistent with.
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