### How *â vs. a* models work

POD (Probability of Detection) is the probability (shown by
the shaded region under the probability density) that a signal, (â, "ahat")
will be larger than the decision threshold. The inset shows POD
plotted on the same size axis at the "*â vs. a*" line. Note the
correspondences at POD = 0.1, 0.5, and 0.9 with the respective shaded
probability densities.

### Reality is Random:

The solid black line is defined as truth. In reality
the truth would be unknown and is to be inferred from the behavior of the
data. The solid black "data" points are observations of signal responses for
a given size. Sometimes they are "censored." Censored
observations are shown as open symbols.

*The figure sequentially presents ten random samples of n=30 and illustrates the magnitude of variability due only to chance.*

The "data" are generated and a censored regression is
performed to produce the most likely model to have given rise to those
observations. That's the blue line. Also shown are the
confidence bounds and prediction bounds (inner and outer dotted lines,
respectively.) Sometimes the blue line (the model) is very close to
the "truth." But sometimes it isn't as can be seen from another random
sample.

In reality we only get to see ONE collection of data, and
from that must estimate the most likely model for the unseen and unknown and
unknowable "truth," and produce its confidence bound that
includes the true a_{90} at least 95 times in every 100 similar
experiments (were we to run the other 99 experiments, which we cannot).