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In this process, definitions of some standard probability density
distributions will be useful.
- The Dirac delta function is such that
 |
(5.26) |
for
, and
 |
(5.27) |
for any
. This probability distribution expresses the
certainty that
.
- The Gaussian probability distribution is
 |
(5.28) |
This probability distribution is a standard default, much beloved
[93,90] of the physics community, for
the distribution over a parameter
, knowledge of which can be
summarized by an expected value
and a degree of variation
about the expected value.
- A perfect top-hat function is
 |
(5.29) |
if
, and
 |
(5.30) |
otherwise. This probability distribution expresses the certainty that
a parameter
lies between
and
, and has equal probability of
lying in any fixed-width subset of that domain. It is useful, for
example, where
is a probability, and therefore must lie between
and
, or where
is a measurement, quantized in units
with an unknown origin by a digital meter, of a parameter
. In the rest of this section, an adapted version of
the perfect top-hat
 |
(5.31) |
if
, and
 |
(5.32) |
otherwise, is used. This distribution has a probability
of being in the top-hat region, and a probability
density outside this region equal to a Gaussian of appropriate width
[95,96] for the probability integrated over this
region to be
. Not only is this a more realistic
belief density than a perfect top-hat, it provides a probability
density gradient outside the top-hat region, which will help the
leapfrog proposal density (section 5.3.4) to reduce the
random walk behaviour of the metropolis method.
Next: Prior Probability Distributions
Up: The Details of the
Previous: The Details of the
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Daniel Christopher Hatton
2004-11-30