Blending seperate probabilities of a joint event

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Tuesday, 19 January 2010
Albert R. Boehm, Retired, Huntsville, AL


by Albert Boehm (retired)



Operational aviation categories such

as below minimums or instrument flight rules are often specified as a combination of low ceiling, Clow and low visibility, Vlow categories Thus we need the probability of the joint event, Pr (Clow, Vlow) both as a forecast probability and a climatic probability

If we have separate probabilities, Pr(Clow and Pro(Vlow),How can they be blended to calculate Pr(Clow,Vlow)?

We note that

if Pr(Clow) and Pr(Vlo) are independent then,


Pr(Clow,Vlo) = I = Pr(ClowPr(Vlow)             (1)

If Pr(Clow and Pr(Vlow) are completely dependent, then

Pr(Clow,Vlow)= D=max[ProClo),Pr(Vlow)]    (2)

These conditions provide useful bounds on Pr(low,Vlow),





 I < Pr(Clow,Vlow) Further we could try an interpolation between the two bounds,

(B)I+(1-B)D +error                                         (4)

wher B was coined the Bilicon facter by Prof. Don Marten, head of the St Louis U. met dept. since a Bilicon is the SLU sports icon

Amazingly when tested on hundreds of climate Pr(Clow,Vlow) for a varity of locations, timrs of day and season with a constant B=0.3, the errer was found to be always less then0.01

The climate errer was easy to determine due to the ready availability of climate summaries,RUSSWOS, The forecast errer is not so easily determined but with no odd verification anomalies, the Bilicon facter has been used successfully used many years now as a standard tool at the Air Force Global Weather Central.



If we hypothesize that Pr(Clow and Pr (Vlow are from a bivariate normal distribution  which

Includes Pr(Clow,Vlow),  then an elegant blending algorithm results.

First  individual “predictors (PC and PV of Pr(Clow) and Pr(Vlow) are determined using the bivariate normal assumption , then PC and PV ar e used to calculate the joint Pr(Clo,Vlow)

If positive correlation is assumed between PC and PV the results similar to the Bilicon facter occurs . But if a negative correlation is used, Pr(Clo,Vlow) can be larger then Pr(Clow) or PrVlow) A remarkable result among the various blending algorithms.