advanced-algorithmic-trading

31
Hence, all we need to do is re-arrange these formulae to provide α and β in terms of µ and
σ. α is given by:
While β is given by:
α =
( 1 − µ
σ 2 − 1 )
µ 2 (3.16)
µ
( ) 1
β = α
µ − 1
(3.17)
Note that we have to be careful here, as we should not specify a σ > 0.289, since this is the
standard deviation of a uniform density (which itself implies no prior belief on any particular
fairness of the coin).
Let’s carry out an example now. Suppose I think the fairness of the coin is around 0.5, but
I’m not particularly certain (hence I have a wider standard deviation). I may specify a standard
deviation of around 0.1. What beta distribution is produced as a result?
Plugging the numbers into the above formulae gives us α = 12 and β = 12 and the beta
distribution in this instance is given in Figure 3.2.
Figure 3.2: A beta distribution with α = 12 and β = 12.
Notice how the peak is centred around 0.5 but that there is significant uncertainty in this
belief, represented by the width of the curve.
3.6 Using Bayes’ Rule to Calculate a Posterior
We are finally in a position to be able to calculate our posterior beliefs using Bayes’ rule.
Bayes’ rule in this instance is given by: