INTERACTIVE MODEL · VOLUME 1, ISSUE 1 · METHODS

The Survey Planner

A live model accompanying “Absence of Evidence”

Everything below runs in your browser with no data sent anywhere. The first model is the textbook case: one detection probability, held constant, and a survey you can watch happen visit by visit. The second lets detection probability differ from site to site, which is where the textbook answer stops being safe.

Model 1: how many blanks do you need?

Set the per-visit detection probability and the confidence you want. The chart draws the false-absence probability \((1-p)^k\) as visits accumulate. Press Run a survey and the model visits a site that really is occupied, one visit at a time, ticking off blanks until it either finds the species or runs out of patience. The posterior probability that the site is occupied updates after every blank, using the prior you set.

Detection probability per visit 0.20 Confidence required 95% Prior belief the site is occupied 0.50 Survey speed

visits needed: 14 visits done: 0 P(blanks so far | occupied): 1.000 posterior P(occupied): 0.500

No survey run yet. The curve is the exact arithmetic; the run is one draw from it.

Model 2: when sites are not alike

Now give each site its own detection probability, drawn once from a Beta distribution with the mean you set and a concentration \(\kappa\) that controls the spread. Small \(\kappa\) means the sites differ a lot: a few are easy, a few are nearly hopeless. The naive planner uses only the mean and answers with \(\ln\alpha / \ln(1-\bar p)\). The honest answer inverts \(B(a,b+k)/B(a,b)\), and the gap between them is what this panel is for.

Mean detection probability 0.20 Concentration kappa 5.0 Confidence required 95% Right-hand panel

SD of p across sites: 0.163 naive visits: 14 visits with heterogeneity: 76 inflation: 5.7x

What this leaves out

Both models assume the site is occupied, that nothing arrives or leaves during the survey, that visits are independent, and that a detection is never a misidentification. Real surveys break all of these. The posterior in Model 1 also needs a prior you had to invent, and the Beta in Model 2 is a convenient shape rather than a measured one. The full argument, including the strongest objection we could construct against our own result, is in the article.