Introduction

CDF maps input between in [0,1]. That is CDF(x) -> (0,1)

Quantile function takes input in (0,1) and return x.

• Not all functionals are invertible.
• Continuous distribution easily satisfies this property
• For discrete distributions we take innfimum of all values [0]

Application in Sampling

• Suppose we want to sample from a given distribution
• We can make a quantile function of it
• Sample uniformly from [0,1], call it p
• x = quatile(p)
• x is the sampled value [1]

Application in point estimation

• Suppose you want to model CTR (click through rate)
• CTR lies in (0,1) and can be presented via beta distribution
• You have clicks and impressions for two ads/items anything
• We are more confident about CTR when you have more impressions
• We can construct a beta distribution with mean as point CTR.
  • ctr = beta(clicks, impression-clicks)
  • Recall in thompson sampling we were increasing alpha and beta by 1
• Now we take 1%, 5%, 10% from quantile function. Call it
• When we have more impressions ctr_x would be close to mean ctr (point estimate), else it will be less
• On side node – this way of constructing distribution gets us away from point estimation and can be used in bayesian approaches

Reference

[0] https://stats.stackexchange.com/questions/212813/help-me-understand-the-quantile-inverse-cdf-function

[1] https://stats.stackexchange.com/questions/184325/how-does-the-inverse-transform-method-work/184337#184337