Good–Turing Unseen Mass

Worked Example

Consider a Skewed population with 6 species A to F, and probabilities [1/2, 1/4, 1/8, 1/16, 1/32, 1/32]. Suppose we draw a "Mixed" sample of size 6: A, A, A, B, B, C.

1. Sample size (N) = 6.
2. Observed species: A, B, C. Total = 3.
3. Unseen species: D, E, F. Actual missing mass = 1/16 + 1/32 + 1/32 = 1/8 (0.125).
4. Singleton species: C appears exactly once. Total singletons = 1.
5. Good–Turing missing mass estimate = Singletons / N = 1 / 6 (0.166666...).

The expected estimator (sum of p_i(1-p_i)^(N-1)) is roughly 0.237, while the expected missing mass after N draws (sum of p_i(1-p_i)^N) is roughly 0.202.

Experiment

In the interactive module, you can test different sampling scenarios:

1. Set Population to "Skewed".
2. Set Sample to "Singletons". The Good-Turing estimate is 1.0, but the actual missing mass is only 0.0625, leading to a large error.
3. Change Sample to "Repeated" (where there are no singletons). The missing mass estimate drops to 0, despite actual missing mass still being positive.

Limitations

The basic Good-Turing estimator calculates missing mass solely based on singleton species. This implies that if a sample contains no singletons, the estimated unseen mass is exactly zero, which is often an underestimation (the actual missing mass remains positive). Furthermore, the estimator does not attempt to distribute this estimated missing mass among specific unobserved species. Note that the standard estimator E[N_1/N] is not strictly unbiased for the same target N missing mass.

References

Good-Turing frequency estimation provides a non-parametric method to estimate the probability of unobserved species.

• A Tutorial on Good-Turing Estimation