Write for the standard normal distribution function. Integrating out the twenty independent past durations gives the threshold-censored normal likelihoodThe frequencies are those in part (a). For a future day, the conditional late-arrival probability is . Conditional independence of the five future durations gives .
Exactly three primitive random coordinates suffice. Draw from the same priors and one independent . Reject if ; otherwise return whether . Conditional on acceptance and the parameters, is uniform on an interval of length ; the true-answer subinterval has length . Reusing this same uniform value is therefore intentional, not an independence mistake.
Indeed the two output contributions at each parameter value areThese are exactly the contributions obtained by integrating the original twenty-five duration noises. Multiplication by the common prior density and normalization therefore give the same joint posterior distribution of . The revised program preserves the query and parameter posterior, but does not retain individual latent durations. This is uniform-interval encoding of conditioning and a binary query.
Here is an executable version, reusing the definitions from part (a). Logarithms avoid multiplying many small probabilities.
def normal_logcdf(z):
value = 0.5 * math.erfc(-z / math.sqrt(2.0))
return math.log(value) if value > 0.0 else -math.inf
def journey_loglik(mu, sigma):
return (sum(n * normal_logcdf((mu - c) / sigma)
for c, n in LATE.items())
+ sum(n * normal_logcdf((c - mu) / sigma)
for c, n in EARLY.items()))
def collapsed_trace(rng):
mu = rng.uniform(0.0, 60.0)
sigma = rng.uniform(1.0, 20.0)
log_likelihood = journey_loglik(mu, sigma)
# This uniform lies in (0, 1], avoiding log(0).
log_u = math.log(1.0 - rng.random())
if log_u >= log_likelihood:
return None
log_h = 5.0 * normal_logcdf((mu - 30.0) / sigma)
return log_u < log_likelihood + log_haccepted_average(collapsed_trace, n, rng) consequently has the same target as the original program. The ideal three-dimensional sample space is with product uniform probability measure, conditioned on the displayed acceptance event. Finite-precision normal tails can underflow to zero in this simple implementation; the mathematical equivalence concerns exact probabilities, and the numerical check in part (d) independently integrates the likelihood in the region carrying appreciable posterior mass. Articles by others on the same topic
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