Treat each observed arrival label as a constraint on an unobserved walking duration. Departures are treated as fixed design information. Writing for the minutes available until work starts, the derived late-label thresholds have frequencies ; the early-label thresholds have frequencies . These are censoring thresholds, not measured journey times. A late label means , and an early label means ; equality has probability zero under the continuous model.
The paper specifies no prior distribution. One explicit proper model iswith independent priors; all times are in minutes. These ranges are a transparent modelling choice, not information deduced from the observations. In particular, this is a prior on the standard deviation , not a uniform prior on the variance . The normal distribution permits negative durations; it is the mathematical approximation requested here, rather than a claim that such durations are physically possible. A truncated model would be a different model.
The following Python implements hard conditioning by rejection sampling.
None marks a rejected trace; an accepted Boolean answers whether all five future arrivals are late. The same parameters are shared across past and future days.import math
import random
LATE = {20: 1, 21: 3, 25: 3, 27: 1, 32: 2}
EARLY = {29: 1, 34: 1, 36: 3, 37: 1, 38: 1, 39: 1, 43: 1, 44: 1}
RECORDS = [(c, True) for c, n in LATE.items() for _ in range(n)]
RECORDS += [(c, False) for c, n in EARLY.items() for _ in range(n)]
def journey_trace(rng):
mu = rng.uniform(0.0, 60.0)
sigma = rng.uniform(1.0, 20.0)
# Twenty latent past durations and five future durations.
past = [rng.gauss(mu, sigma) for _ in RECORDS]
future = [rng.gauss(mu, sigma) for _ in range(5)]
if not all(t > c if late else t < c
for t, (c, late) in zip(past, RECORDS)):
return None
return all(t > 30.0 for t in future)
def accepted_average(trace, n, rng):
if n <= 0:
raise ValueError("n must be positive")
successes = accepted = 0
while accepted < n:
result = trace(rng)
if result is not None:
accepted += 1
successes += int(result)
return successes / nFor example,
accepted_average(journey_trace, n, random.Random(2021)) estimates the desired conditional probability from accepted traces. Accepting False is essential: treating a false answer as a rejected trace would condition on the desired conclusion. The model deliberately emphasizes clear semantics; rejection can be inefficient because all twenty past-label constraints must hold together. Articles by others on the same topic
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