Stochastic urn simulation of the two-region feedback loop

Simulates the patrol-and-discovery process step by step (C++ kernel). Each step the patrol visits A with probability equal to the current share and discovers a crime there with probability equal to that region's true rate; the naive update adds the discovery to the count, the corrected update adds it weighted by one over the presence that produced it. Public reports, when rho > 0, arrive from a region in proportion to the true rates regardless of presence.

Usage

morie_feedback_loop_sim(
  lam_a,
  lam_b,
  c_a0,
  c_b0,
  n_steps = 1000L,
  update = c("naive", "corrected"),
  rho = 0,
  n_sims = 100L,
  seed = 0
)

Arguments

lam_a, lam_b

True crime rates (per patrol visit) of regions A and B; positive.

c_a0, c_b0

Initial discovered-crime counts; positive.

n_steps

Number of steps to iterate.

update

"naive" (discovered counts fed back as they are) or "corrected" (each discovery discounted by the presence that produced it, so the increment is the true rate).

rho

Share of crimes that reach the record through public reports independent of patrol presence, in [0, 1]. With rho > 0 the naive share is proved never to exceed \(\max(x_0, \lambda_A/(\lambda_A + \rho\lambda_B))\) (Research.P4.rho_cap): reports cap the loop strictly below 1 but strictly above the true-rate proportion unless rho = 1.

n_sims

Number of independent runs.

seed

Philox seed (non-negative integer). Run i draws its uniforms from stream i - 1 of .morie_random_uniform, three per step when rho > 0 and two otherwise, so the R and Python arms produce identical paths for the same seed.

Value

A list with share_a (an n_sims by n_steps + 1 matrix of share paths), final (the final shares) and limit (the proved mean-field limit).

Examples

s <- morie_feedback_loop_sim(0.3, 0.2, 10, 10, n_steps = 2000, n_sims = 20, seed = 1)
stats::median(s$final)     # near 1
#> [1] 0.7627981
s$limit$theorem
#> [1] "Research.P4.naiveShare_tendsto_one"