Bayesian Aim

Calculates the probability of clicking circles

Concept

Any osu beatmap perscribes a series of cursor movements. We want to measure how difficult these movements are. Historically, difficulty splits into two components: agility and precision. Here we combine both of these aspects into one computation.

We describe a range of comfortable cursor movements. Some of these movements fit a beatmap. We can rephrase the agility-precision split as follows:

• Agility: how uncomfortable are the prescribed cursor movements?
• Precision: what margin of error does the beatmap permit?

The answer to both of these questions is probability.

We start with a model of cursor movement. This model assigns different probabilities to various movements. Comfortable movements have high probabilities; uncomfortable movements have low probabilities. A beatmap prescribes some range of movements. We add up the total probability of this range under our model.

This total probability measures both agility and precision. A more agile, less comfortable, range of motion has a lower probability. A more precise, narrower, range of motion also has a lower probability.

See also the included math-blog-posts. These more fully describe the above concepts.

Roadmap

Document underlying math:

• jerk as white noise
• integrals as linear transformations
• Bayes' law
• the failures of normal distributions
• the actual distribution in use

Answer any questions with supporting documentation. Iterate until no more questions.

Agile development:

• write psuedocode
• write unit-tests
• write code

Minimum feature list:

• read .osu files (library already exists)
• evaluate difficulty
  • most likely, difficulty will be evaluated per-segment, with some other algorithm aggregating the segments.
• produce sample cursor paths
  • this lets us visually check that our model is correct
  • ideally, this would take the form of a generated .npy file
  • however, a .gif would also suffice

Deployment and monitoring (i.e., parameter tuning):

• some parameters have objective mathematical descriptions
  • analytical derivations will work for some
  • computational methods might work for others
  • one of the parameters has 100+ dimensions, so that'll be fun
• other parameters should be fit to human player data
  • that's what you guys do best, right?