The estimator
from zero.

Think of the library as a room with a door. Every minute, each person inside flips a biased coin and leaves with probability μ. New people walk in at random, more at noon than at seven in the morning. That is the whole model of how the count changes, and it is the only place time enters.

Try this: set N_t to 40 and move the hour from 07:00 to 13:00. Watch the bars shift right by the arrivals. Then make the stay short and watch the dashed survivors curve slide left.

A body in the room moves some of the echoes the receiver adds up, so the channel wobbles more. The instrument measures that wobble as the spread of the ratio between a receiver's two antenna chains, takes its logarithm, subtracts the receiver's empty-room floor, and keeps the median over the receivers that report: the panel level L. The model says how high L tends to be for each headcount, and how far one minute can stray.

Try this: compare how far the curve rises between 0 and 10 people with how far it rises between 80 and 120. Then drag the observed value far above the band, once with the outlier component and once without.

Phones, watches and laptops advertise over Bluetooth under addresses that change every few minutes. The fleet counts the distinct non-public addresses heard in a minute, keeping only those strong enough (−70 dBm at some receiver) to be near the floor: B. More people means more addresses, but nobody knows how many addresses one person adds. That number is α.

Try this: keep B at 224, the busiest minute of the development day in the thesis figure, and move α from 8 to 2 to 0.5. The same minute means 28, then 112, then more people than the room holds.

Imagine two libraries. In one, 30 people each carry devices worth 2 addresses. In the other, 60 people each carry devices worth 1 address. Both rooms produce the same Bluetooth count, and if the CSI curve bends at twice the count in the second, the same CSI level too. No radio can tell them apart. This is the central difficulty of the thesis, and it has a name in sensor networks: blind calibration of a global gain.

Try this: move κ anywhere. The two Bluetooth distributions stay on top of each other. Then read the table below the plot: the count process is where the two worlds differ, and the capacity bound is where one of them stops existing.

If the data cannot choose α, someone has to state what is plausible, before any count is opened, and the estimator has to carry that uncertainty instead of hiding it. The thesis declares two priors. A is log-normal around 2 addresses per person; B spreads evenly on a log scale between 0.7 and 6. The estimator runs one filter per quadrature node, seven values of α, and mixes their answers with the prior weights.

Try this: leave the priors frozen and move the excess addresses. The dots spread over a factor of about ten under A and about seven under B, whatever the excess: that width belongs to the prior, and no amount of data narrows it. Then compare the browser's nodes with the server's in the table.

Here everything meets. At midnight the estimator knows nothing: every count from 0 to 160 is equally likely. Each minute it does three things. It predicts: pushes yesterday's belief through the queue of lesson 1. It weighs: multiplies each count's probability by how well that count explains what the radios heard (lessons 2 and 3). It normalises: rescales so the probabilities sum to one. That is all a hidden Markov model filter does.

Try this: scrub to a morning minute and watch the dashed prediction, the blue likelihood and the posterior bars. Switch the Bluetooth count off and see how much wider the answer gets. Then run the full mixture: your browser reproduces the posterior that the frozen Python code wrote for this day.

Run the filter on another exported day: 09-30 · 10-01 · 10-02 · 10-03 · 10-04 · 10-05 · 10-06 · 10-07 · 10-08 · 10-09 · 10-10 · 10-11 · 10-12 · 10-13 · 10-14 · 10-15.

The arrival table and the CSI curves were never fitted to a people count, because the thesis has none to give them. They come from the radios alone, by expectation–maximisation: guess the parameters, ask the filter (run forwards and backwards) how many people were probably present each minute, refit the parameters to those soft guesses, and repeat. Each round makes the observed data at least as probable as before, for the exact EM steps.

Try this: read the evidence column: the first round gains thousands of nats, the eighth a fraction of one. Then click the heat table to compare a Monday noon with a weekend noon.

A posterior over 161 counts is honest but hard to read. The estimator reads it out in three ways, all from the same distribution, so they cannot contradict each other. Three levels answer "anyone here, a few, or busy?". Six bands answer "roughly how many?". And the 90 % band set says how wide the honest answer has to be.

Try this: with your own shape, widen the spread until the set grows from one band to four. Then switch to the frozen model's day and scrub from night to noon: the levels sharpen long before the bands do.

A forecast that says "70 % busy" is neither right nor wrong on one minute. It is judged by a score that rewards putting probability where the outcome lands, and punishes near misses less than far ones. The thesis uses the ranked probability score for the levels (H2) and coverage together with width for the band sets (H3), with the date as the unit, and a test that makes no large-sample assumption.

Try this: in the second panel, move the truth and watch the minimum of the expected score follow it: lying about your belief can only cost you. In the third, change the illustrative differences until the test stops holding.

Every CSI level is measured above a floor: what that receiver reads in an empty room. The floor is not fixed. On this fleet single receivers jump between empty nights, by more than 0.8 on a natural-log scale on about one receiver-night in five, each on its own nights. The estimator uses the median over receivers, and a median ignores a minority. H1 asks how much that buys.

Try this: raise the jump chance until the panel row starts to flare too. Then press "Repeat 400 times" and compare the spread of simulated panel ratios with the measured 0.103.

Every model is a set of convenient lies, and a thesis has to say which ones it tells. Three matter here. Minutes are treated as independent given the count, and they are not. The Bluetooth count is modelled as at least as variable as a Poisson count, and it is often less. And the CSI floor is assumed stable between mornings, which lesson 10 showed holds for the panel and fails for single receivers.

Try this: set ρ to 0.35, the measured value for the Bluetooth count: an hour of minutes carries the evidence of about 29 independent ones. Then set the variance-to-mean ratio to 0.5 and compare the shapes.

See the frozen model's output on every exported day: the frozen estimator, day by day. For how one minute of CSI is made before any of this, how a level is made.

The model, its symbols and every measured number come from the thesis chapters "Research Hypotheses and Evaluation Criteria", "A Commodity Multi-Link Sensing Instrument" and "A Fused Occupancy Estimator", and from the frozen model file (notebooks/studies/2026-10-09-fusion-estimator/results/frozen-model.json, copied into the vault by notebooks/studies/2026-10-09-fusion-estimator/export_site.py). Literature is cited by first author and year as in the thesis. Lab assumptions that are not the thesis's are labelled where they appear: the illustrative arrival rate of lesson 4, the wobble without a jump in lesson 10, and the twelve per-date differences of lesson 9, which are not results.