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In the context of estimating geophysical quantities from remotely sensed data (inverse theory), what do the terms ex ante and ex post mean? For context, see for example this paper by T. von Clarmann (emphasis mine):

The actual value of $χ^2$ allows to conclude if the differences $\mathbf{\hat{x}}_\textrm{val} - \mathbf{\hat{x}}_\textrm{ref}$ are consistent with the ex ante estimates of the uncertainties of the difference, represented by its covariance matrix $\mathbf{S}_\textrm{diff}$ , or if there is a significant inconsistency.

and:

Instead, we use independent reference measurements, which allow to infer ex post estimates of bias, precision, and accuracy.

Is there any difference between the terms ex ante and a priori?
Likewise, is there a difference between ex post and a posteriori?
If yes, what are those differences?

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Ex Ante means before the event. Ex Post means after the event. In this example, I think this means before and after the event that gives the statistical difference you're testing, respectively.

On the other hand, a priori and a posteriori are terms from philosophy, respectively denoting knowledge that is logically derived, and knowledge that requires empirical evidence. (Wikipedia)

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    $\begingroup$ What event would that be? And a priori and a posteriori do have a rather specific meaning in (Bayesian) statistics too. $\endgroup$
    – gerrit
    Jul 23, 2014 at 23:09
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"Ex ante" & "ex post" are used in forecasting:

An ex ante forecast uses only the information available at a given time to predict what'll happen after that time.

An ex post forecast also uses information available after the given time.

So while an ex post sales forecast based on GDP growth in a region may be informative, the ex ante sales forecast for practical application will need to be based on forecasts of future GDP growth.

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To me, there is a meaning of process in the term a priori: a priori, I think so, but my mind might change through the process of understanding of the new ideas. However, ex ante refers to a moment that is right before an event: ex ante Wall Street fell down, I sold my assets.

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