WOLFRAM|DEMONSTRATIONS PROJECT

Comparing Ambiguous Inferences when Probabilities Are Imprecise

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total count •
100
1000
10000
your selection of natural frequencies
(bolder color)
base rate ▲
0.2
sensitivity ●
0.8
specificity ●
0.7
benchmark (BM) choices
(lighter shading)
BM base rate ▲
0.2
BM sensitivity ●
0.8
BM specificity ●
0.7
Truth Table and Natural Frequencies
True
Positive
False
Negative
False
Positive
True
Negative
S
1
1
0
0
D
1
0
1
0
--
--
--
--
Frequency
16
4
24
56
total count 100
Graphics of Inverse Probabilities ■
How do you interpret the result of the diagnostic test for the level of a state variable when some or all of the information underlying the inference is ambiguous (imprecise)?
Let
S
be the logical truth value (1 or 0) of a proposition about the state variable (e.g. a disease is present or absent, or a failure is recorded or not) and let
D
be the logical truth value of a proposition about the outcome of an imperfect diagnostic test being a positive indicator for the state (e.g. a blood test result for this disease, a quality control check for a manufacturing failure). From a statistical perspective there are three precise numerical inputs that feed into a coherent posterior inference about binary-valued
S
after having observed the result of the binary-valued diagnostic signal
D
: a sensitivity number, a specificity number, and a base rate number (as explained in the Details section). The sliders on the left control these three numbers, and the table and graphical representation update dynamically. To facilitate the "what-if" exploration of the effects on posterior inferences of ambiguities (imprecision) in sensitivity, specificity, and base rate information, there are two sets of sliders: the lower set, a benchmark, remains slightly faded out in the picture as the upper set of slider values varies.