Exact law of the stochastic two-region urn with equal rates
Source:R/feedback_loop.R
morie_feedback_loop_urn_law.RdOne discovery per step, credited to region A with probability equal to
the current share of A and adding one count there. The share is a
martingale for any reinforcement (Research.P4.Urn.urn_step_martingale).
Started from one count each, the number of discoveries credited to A
after n_steps draws is exactly uniform on 0:n_steps
(Research.P4.Urn.polya_uniform), so the share
\((1 + j)/(n + 2)\) is uniform on its grid and the probability that
it lies in \([1/4, 3/4]\) is at least \(1/4\) at every horizon
\(n \ge 2\) (Research.P4.Urn.polya_no_concentration).
Value
A data frame with j (discoveries credited to A),
share_a and prob, with attributes "prob_middle"
(probability that the share lies in \([1/4, 3/4]\)) and
"theorems".
Details
The mean-field recursion with equal rates keeps the share at one half forever; the stochastic process locks in early luck and converges to a random limit that is uniform on (0, 1). "Equal rates, therefore no runaway" is false for the process a department actually runs.
See also
morie_feedback_loop_sim for the simulated urn with
unequal rates.
Examples
law <- morie_feedback_loop_urn_law(10)
law$prob # all 1/11
#> [1] 0.09090909 0.09090909 0.09090909 0.09090909 0.09090909 0.09090909
#> [7] 0.09090909 0.09090909 0.09090909 0.09090909 0.09090909
attr(law, "prob_middle") # at least 1/4
#> [1] 0.6363636