Michael Halls Moore's Advanced Algorithmic Trading PDF
By Michael Halls Moore
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Well, these two concepts neatly correspond to the mean and the variance of the beta distribution. Hence, if we can find a relationship between these two values and the α and β parameters, we can more easily specify our beliefs. 15) 31 Hence, all we need to do is re-arrange these formulae to provide α and β in terms of µ and σ. 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.
It states that we have equal belief in all values of θ representing the fairness of the coin. The next panel shows 2 trials carried out and they both come up heads. Our Bayesian procedure using the conjugate Beta distributions now allows us to update to a posterior density. Notice how the weight of the density is now shifted to the right hand side of the chart. This indicates that our prior belief of equal likelihood of fairness of the coin, coupled with 2 new data points, leads us to believe that the coin is more likely to be unfair (biased towards heads) than it is tails.
1 The Bayesian Approach While we motivated the concept of Bayesian statistics in the previous chapter, I want to outline first how our analysis will proceed. This will motivate the following sections and give you a "bird’s eye view" of what the Bayesian approach is all about. As we stated above, our goal is estimate the fairness of a coin. Once we have an estimate for the fairness, we can use this to predict the number of future coin flips that will come up heads. We will learn about specific techniques as we cover the following steps: 1.
Advanced Algorithmic Trading by Michael Halls Moore