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Alpha, Beta, Gamma…

\alpha, \beta, \gamma

I made the following  notes in reading John Cook’s post on the rule of three.

Notes:

  • \lim _{n\rightarrow \infty} (1-\frac{3}{n})^n =e^{-3}\approx 0.05
  • Bayes theorem: f(\theta|D)=\frac{f(D|\theta) f(\theta)}{\int_\theta f(D|\theta)f(\theta)\, d\theta}
  • Binomial distribution (N trials, s successes) probability density function:f(D|\theta)=^N C_s \theta ^s (1-\theta)^{N-s}
  • Beta distribution probability density function: f(t;\alpha ,\beta)= \frac{1}{B(\alpha ,\beta)}t^{\alpha -1}(1-t)^{\beta-1} where B(\alpha , \beta)=\int_0^1 t^{\alpha -1}(1-t)^{\beta -1} \, dt
  • In estimating the rate of an event that hasn’t happened  yet using Bayes theorem, let the event  be modeled by binomial distribution, and consider the prior distribution for the probability of event assumed to be a beta distribution(\alpha , \beta). To get the uninformative prior that is used in the post we will just let \alpha =\beta =1. Let \theta be the parameter of interest.
  • With these assumptions:

f(\theta | N, s) =\frac {\theta ^s (1 -\theta)^{N-s} \theta ^{\alpha -1}(1 - \theta)^{\beta -1}}{\int_0^1 \theta ^s (1-\theta)^{N-s} \theta ^ {\alpha -1} (1 - \theta) ^{\beta -1} d \theta }

Simplifying yields:

f(\theta | N, s)=\frac{\theta^{s +\alpha -1} (1 -\theta)^{N -s +\beta -1} }{B( s +\alpha , N - s +\beta)}

This is a beta distribution ( s + \alpha, N -s +\beta).  Setting \alpha = \beta =1  and s =0 yields the beta distribution (1, N+1) as indicated by the post.

The following graphic illustrates the effect of changing N on the posterior probability.

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Categories: Mathematica, Mathematics
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