advanced-algorithmic-trading

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Figure 3.3: The prior and posterior belief distributions about the fairness θ.
µ post =
α
α + β
(3.24)
=
22
22 + 52
(3.25)
= 0.297 (3.26)
(3.27)
While the standard deviation σ post is given by:
σ post =
=
√
αβ
(α + β) 2 (α + β + 1)
√
22 × 52
(22 + 52) 2 (22 + 52 + 1)
(3.28)
(3.29)
= 0.053 (3.30)
In particular the mean has sifted to approximately 0.3, while the standard deviation (s.d.)
has halved to approximately 0.05. A mean of θ = 0.3 states that approximately 30% of the time,
the coin will come up heads, while 70% of the time it will come up tails. The s.d. of 0.05 means
that while we are more certain in this estimate than before, we are still somewhat uncertain
about this 30% value.
If we were to carry out more coin flips, the s.d.
would reduce even further as α and β
continued to increase, representing our continued increase in certainty as more trials are carried
out.
Note in particular that we can use a posterior beta distribution as a prior distribution in a
new Bayesian updating procedure. This is another extremely useful benefit of using conjugate
priors to model our beliefs.