Standard Probability Distributions

In this process, definitions of some standard probability density distributions will be useful.

• The Dirac delta function is such that
  

  $$\delta{}(x) = 0\textrm{,}$$ (5.26)

  
  
  for $x \neq{} 0$, and
  

  $$\int_{-\epsilon}^{\epsilon}\delta{}(x)\mathrm{d}x = 1\textrm{,}$$ (5.27)

  
  
  for any $\epsilon > 0$. This probability distribution expresses the certainty that $x = 0$.
• The Gaussian probability distribution is
  

  $$G(x; \mu{}, \sigma{}) = \frac{1}{\sqrt{2\pi\sigma^2}}\exp{}\left(-\frac{(x-\mu{})^2}{2\sigma^2}\right)\textrm{.}$$ (5.28)

  
  
  This probability distribution is a standard default, much beloved [93,90] of the physics community, for the distribution over a parameter $x$, knowledge of which can be summarized by an expected value $\mu$ and a degree of variation $\sigma$ about the expected value.
• A perfect top-hat function is
  

  $$T_0(x; l, h) = \frac{1}{h-l}\textrm{,}$$ (5.29)

  
  
  if $l \leq{} x \leq{} h$, and
  

  $$T_0(x; l, h) = 0\textrm{,}$$ (5.30)

  
  
  otherwise. This probability distribution expresses the certainty that a parameter $x$ lies between $l$ and $h$, and has equal probability of lying in any fixed-width subset of that domain. It is useful, for example, where $x$ is a probability, and therefore must lie between $0$ and $1$, or where $x$ is a measurement, quantized in units $h-l$ with an unknown origin by a digital meter, of a parameter $\frac{h+l}{2}$. In the rest of this section, an adapted version of the perfect top-hat
  

  $$T(x; l, h)\frac{127}{128(h-l)}\textrm{,}$$ (5.31)

  
  
  if $l \leq{} x \leq{} h$, and
  

  $$T(x; l, h) = G\left(x; \frac{h+l}{2}, \frac{h-l}{5.32}\right)\textrm{,}$$ (5.32)

  
  
  otherwise, is used. This distribution has a probability $\frac{127}{128}$ 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 $\frac{1}{128}$. 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.