...this new knowledge [a specific order for 40 green widgets], which makes the problem so hard for our common sense, causes no difficulty at all in the mathematics. The previous equations still apply, with the sole difference that the stock of green widgets is reduced from 50 to 10.
The above seems intuitively plausible, but let's follow Jaynes's advice to always carefully derive results from the basic laws of probability theory, rather than making intuitive leaps. The new information for Stage 4 is a proposition instead of an expected value for the prior distribution to satisfy; the proper procedure then is to take the prior distribution of Stage 3 and condition on to obtain the Stage 4 distribution. Let's do that now.
Recall that is the number of orders for red widgets,
is the number of orders for yellow widgets, and
is the number of orders for green widgets. From (14.72), the Stage 3 prior
distribution factors into independent distributions for the sets of variables
, , and :
From (14.69), (14.70), and (14.71), we also see that factors into independent distributions for each variable :
Thus, conditioning on affects only the distribution for . Furthermore, the distribution for is an exponential distribution, and as such has the easily verified general property that for any ,
Using , Jaynes's assertion follows directly.