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Chapter 14: Simple applications of decision theory
 p. 428, equation (14.9): This isn't quite stated correctly. We cannot
have
for all propositions ; consider, for
example, defining
. A correct statement might require
that be a proposition asserting particular values for model variables
,
; that be a proposition asserting a particular
value for a model variable , distinct from the variables ; and
that be a proposition asserting a particular value for another model
variable , distinct from and the variables .
 p. 429, Theorem: This isn't stated quite correctly. The fact that is a
possible decision, given , does not imply that
. Is
meant as an extra condition? Furthermore, in
equation (14.14), given that
, the implication
holds, but the implication holds only if
.
 p. 433, equation (14.32), first line: ``
''
should be ``
''.
 p. 440, first full paragraph: ``Woodword'' should be
``Woodward.''
 p. 444, first line: the reference to (11.46) should be (11.48).
 p. 447, equation (14.79): I believe the variable r on the righthand side
of the equation should be omitted, to give a numerator of
.
 p. 447, equation (14.82): ``
''
should be ``
''.
 p. 448, equation (14.83):
``
'' should
be ``
''.
Jaynes states,
...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.
