The conjugate Bayesian estimator statistical algorithm

Introduction

This repo contains a reference implementation for a statistical algorithm called Conjugate Beta Estimator (CBE) for computing confidence intervals (CIs) for population means using (weighted) sample means and potentially noisy labels.

Basic algorithm

The basic formula for CBE is

alpha = mu*n + alpha_prior
beta = (1-mu)*n + beta_prior
ci = [ppf(0.05, alpha, beta), ppf(0.95, alpha, beta)]

where mu is the mean of the (weighted) sample labels, n is the sample size in bits, and ppf is the inverse CDF function.

Consider using the Kumaraswamy distribution instead of Beta for a more efficient analytical inverse CDF function.

CBE is an unbiased estimator of the population mean because CBE's mean is alpha/(alpha + beta). As n->inf, alpha_prior and beta_prior wash out, and the expression simplifies to mu.

The idea behind CBE -- a distribution whose mean converges to the population mean and whose variance shrinks with the number of labels -- can be generalized to non-Beta distributions if you need a different support than [0,1]. For example, the for support on [0, inf] you can use a Gamme distribution that's parameterized as

Gamma(alpha=1+(n*mu), beta=1+n)

For support on [-inf, inf] the Gaussian distribution can be parameterized as

Gaussian(mean=(prior + n*mu)/(n+1), var=1/(n+1))

CBE can be applied in streaming applications, where mu can represent a moving average.

Accounting for label noise

Both mu and n can be adjusted to account for label noise.

mu should be adjusted using the Rogan Gladen (RG) estimator for the sample mean:

mu_modified = (mu + specificity - 1) / (sensitivity + specificity - 1)

n should be adjusted using the following formula:

num_bits_per_label = (1 - entropy((sensitivity + specificity) / 2))
n_modified = num_bits_per_label * n

the accuracy adjusted CBE formula is therefore

alpha_modified = mu_modified*n_modified + alpha_prior
beta_modified = (1-mu_modified)*n_modified + beta_prior
ci = [ppf(0.05, alpha_modified, beta_modified), ppf(0.95, alpha_modified, beta_modified)

The reason for the num_bits_per_label formula is that the rg formula is increasingly unstable when the mean of sensitivity and specificity approaches 0.5 (the max entropy value) and the rg denominator approaches 0. The rg formula is maximally stable when sensitivity = specificity = 1, which is the case when labels are perfectly accurate. Therefore, we want the CI derived from the Beta distribution to grow wider as (sensitivity + specificity)/2 approaches 0.5 from 0 or 1.

The relationship between average accuracy and # of bits per label is visualized in the graph below.

Mathematical derivation

The following derivation shows that the 1-H(X) formula, where H is entropy and X is (sensitivity + specificity)/2, can be expressed as the Bayesian information gain, or KL divergence between the posterior P and uniform prior Q:

$$D_{\text{KL}}(P \parallel Q) = p * \log_2\left(\frac{p}{q}\right) + (1-p) *\log_2\left(\frac{1-p}{1-q}\right) =$$ $$p *\log_2\left(\frac{p}{0.5}\right) + (1-p) * \log_2\left(\frac{1-p}{0.5}\right) =$$ $$p *\left(\log_2(p) - \log_2(0.5)\right) + (1-p) * \left(\log_2(1-p) - \log_2(0.5)\right) =$$ $$p *\log_2(p) + p + (1-p) *\log_2(1-p) + (1-p) =$$ $$p *\log_2(p) + (1-p) * \log_2(1-p) + 1 = $$ $$ 1 - H(X) $$

Future Directions

If you want to introduce uncertainty about sensitivity and specificity, it might be possible to mulitply num_bits_per_label by another 1-H(X) term, where X is the entropy of the sensitivity or specificity distribution. The intuition is that an infinite number of labels whose accuracy is unknown (uniformly distributed on [0,1]) are as informative as zero labels.

License

Conjugate Estimators is MIT licensed, as found in the LICENSE file.