AscentStats
Bayesian model for IFSC bouldering competitions
These ratings are estimated by a Bayesian statistical model fitted in Stan, using results from IFSC boulder competitions. Each climber and each boulder problem has an underlying latent grade. We estimate the grades from the observed pattern of successes and failures across hundreds of routes and thousands of climbers per season.
The success model
For climber $c$ on route $r$, the probability of success on a single ascent attempt is
\[P(\mathrm{success}) = \mathrm{logistic}\bigl(m \cdot (\mathrm{climberGrade}_c - \mathrm{routeGrade}_r)\bigr)\]where $m = \log 2$ controls how sharply skill differences translate into win probability. A 1-unit grade gap is roughly a 2:1 success ratio.
Each boulder contributes two observations: did the climber reach the zone hold (the easier sub-game) and did they reach the top (the harder game). Zones and tops are modelled as separate routes, with their own grades.
Climber priors
Each climber’s grade has a Normal prior centred on their previous-season estimate. New climbers start with a neutral prior. Ratings of climbers who skip a season decay by 1% per year of inactivity, so a climber who last competed three years ago retains 97% of their prior strength.
Route priors and event tiers
Route grades are themselves drawn from population-level distributions:
\[\mu_\mathrm{top} \sim \mathrm{Normal}(0, 2),\qquad \mu_\mathrm{zone} \sim \mathrm{Normal}(-2, 2)\] \[\mathrm{routeGrade}_r \sim \mathrm{Normal}(\mu_{\cdot} + \mathrm{adj}_r,\ 2)\]The offset $\mathrm{adj}_r$ depends on the event tier of the route’s event:
| Tier | Events | Offset |
|---|---|---|
| 1 | World Cup / World Climbing Series, World Championships, Olympics | 0 (reference) |
| 2 | European Cup / Europe Series | $\delta_\mathrm{euro} \sim \mathrm{Normal}(0, 1)$ |
| 3 | Continental Cup / Continental Championships | $\delta_\mathrm{other} \sim \mathrm{Normal}(-0.5, 1)$ |
The tier-2 and tier-3 offsets are estimated jointly with everything else. This lets the model learn from the data that, for instance, European Series routes tend to be set easier than World Cup routes (in part because the world’s strongest climbers don’t show up to set the upper bound of skill). The posterior for these offsets is saved each season alongside the climber and route results.
Why this matters
Without an event-tier offset, a climber who only competes at lower-tier events would appear in the same skill distribution as those competing on World Cup walls. The tier offset corrects for that — a climber dominating a development circuit ends up rated according to development-circuit difficulty, not World Cup difficulty.
Other features
- Group-uncertainty handling: when a qualification round splits climbers into groups A/B without recording the assignment, the model marginalises over both possible group orderings rather than guessing.
- Logistic likelihood: the climb-or-no-climb outcome of each attempt is modelled with a Bernoulli-logit, so the data drives both climber and route grades simultaneously.
- Stan / HMC: posterior inference uses 4 chains, providing 95% credible intervals for every climber and route.