Probability is interchangeable with proportion; although one is theoretical, the other arithmetical.
Note: Mathematical theories go complicated very fast; in fact, the description of random processes is already impractical and requires too much information. To put the mathematical model into practical use, vast simplification is needed.
(Ref: Scholtz, P266.)
Nassim Nicholas Taleb, Skin in the game. "Ergodicity is not statistically identifiable, not observable, and there is no test for time series that gives ergodicity, similar to Dickey-Fuller for stationarity (or Phillips-Perron for integration order)." "If your result is obtained from the observation of a time series, how can you make claims about the ensemble probability measure?"
Fourier Transforms, Z-transform
Table 1: Interpretations of probability
|Ontic (实存) probability||Epistemic (认识) probability|
|long run frequency||objective degree of belief|
|single-case propensity||subjective degree of belief|