Chapter 18: The distribution and rule of succession
- p. 563, eqn. (18.22): should be .
- p. 570, eqn. (18.39):
( instead of ).
- p. 576, second paragraph, end of line six: ``existance''
should be ``existence.''
- p. 576, second paragraph: ``As we saw earlier in this chapter, even the
and in Laplace's formula turn up when the `frequentist' refines his
methods...'' Actually, this is discussed later in the chapter --
see eqn. (18.68).
- p. 577, text preceding equation (18.58): The reference to equation
(18.55) should probably be (18.56).
- p. 579, last paragraph of section 18.15, line six: ``thoery''
should be ``theory.''
- p. 580, first line: should be .
- p. 580, third line after (18.69): ``Pearson and Clopper''
should be ``Clopper and Pearson.''
- p. 581, third line from bottom: should be .
- p. 582, eqn. (18.73): should be ; also, in the second line,
first factor, should be .
- p. 582, first three lines after (18.73): should be in each
- p. 582, eqn. (18.76): should be .
- p. 583, eqn. (18.78), second line: should be .
- p. 583, eqn. (18.79):
and the factor
- p. 583, eqns. (18.80), (18.81), and (18.82): should be .
- p. 584, second full paragraph, lines 3 and 6: should be .
- p. 584, second full paragraph, end of line 12, and also line 13:
should be ``uncertainty.''
- p. 586, eqn. (18.87):
- p. 587, first line after (18.93): ``If we substitute (18.93)...''
Should this be (18.91)?
- p. 554, eqn. (18.1): This definition cannot hold true for arbitrary
propositions ; for example, what if implies ? This kind of problem
occurs throughout the chapter. I don't think you can really discuss the
distribution properly without explicitly introducing the notion of a sample
space and organizing one's information about the sample space as a graphical
model in which has a single parent variable , with defined
as the proposition . For those unfamiliar with graphical models /
Bayesian networks, I recommend the following book:
- p. 555, eqn. (18.3): This appears to be at odds with Chapter 12, which
advocates the improper Haldane prior (proportional to
describing the ``completely ignorant population.'' However, that chapter also
argues that the Haldane prior applies when one does not even know whether or
not both outcomes are possible...and that the uniform prior applies if one
does know that both outcomes are possible. (I argue in my comments on
Chapter 12 that the uniform prior is the correct ignorance prior in general
- p. 555, eqn. (18.7): For those who may be confused by this equation,
, not the
probability density of given .
- p. 556, third line after (18.9): ``But suppose that, for a given ,
(18.8) holds independently of what might be; call this `strong
irrelevance.' '' If (18.8) holds for any proposition , then in
particular it holds for the proposition
from (18.8) we have
then since implies
, we also have
. Thus, this definition of ``strong irrelevance''
actually ensures that is highly relevant to . As before, this
discussion really needs to be rewritten in terms of graphical models to get it
right, in particular making use of the notion of -separation.
- p. 583, eqn. (18.78): To get the second line from the first, use these
- For the binomial distribution, and
- p. 586, third full paragraph, first sentence: ``An important theorem of
def Finetti (1937) asserts that the converse is also true:...'' What Jaynes
says here is not true for finite ; it only holds in the limit as
. As a counterexample, consider draws without replacement
from an urn containing balls, with black and white. The
sequence of draws
is exchangeable, but
cannot be generated by any distribution.
To see this, note that once we know the values of
know the value of with certainty, because we know the total number of
balls of each color in the urn.
- p. 586, third full paragraph, second sentence: Even in the limit
, for this statement to be true in general we must allow
to be a generalized function--that is, we must be able to assign
nonzero probability mass to single points using delta functions.
- p. 587, second sentence after (18.89): For this sentence to be true
requires that matrix be nonsingular, where
To see that is in fact nonsingular, note that for and
. Then for arbitrary one can solve for in by backsubstitution.