Markov Model Probability of FC

[peterhusisian]

Calculating the joint distribution over possible player cursor positions given the placements and times of notes in a beatmap is prohibitively expensive. It relies on a product of multiple conditional probabilities, where the number of elements conditioned on increases linearly with the number of notes in the beatmap. A Markov Model mitigates the increasing number of parameters of the conditional distributions by instead assuming that only the k most recent previous elements play the largest role in the probability in comparison to those before them. The aim is to approximate the exact joint distribution of cursor positions given the beatmap using a Markov Model, then train the parameters of the conditional distribution used to form this approximation through Maximum Likelihood Estimation on beatmap replay data. Attached is a short writeup going into further detail of a proof of concept:

PP_Rework (2).pdf

[peterhusisian]

I realized that making a distribution over cursor positions rather than over the joint distribution of h_i's, where h_i is whether the ith note was hit, was overly contrived. I'm working on rewriting this in a much simpler fashion that removes as much fluff as possible that also factors in a distribution over a skill vector.

[peterhusisian]

This approach is significantly less contrived but still leverages a Markov Model, but this time it factors in a skill distribution:
Markov_Model_Skill_Distribution_PP_Rework (1).pdf