By Michael Halls Moore
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13) Notice that the term on the right hand side of the proportionality sign has the same form as our prior (up to a normalising constant). 3 Multiple Ways to Specify a Beta Prior At this stage we’ve discussed the fact that we want to use a beta distribution in order to specify our prior beliefs about the fairness of the coin. However, we only have two parameters to play with, namely α and β. How do these two parameters correspond to our more intuitive sense of "likely fairness" and "uncertainty in fairness"?
3: The prior and posterior belief distributions about the fairness θ. 05. 3 states that approximately 30% of the time, the coin will come up heads, while 70% of the time it will come up tails. d. 05 means that while we are more certain in this estimate than before, we are still somewhat uncertain about this 30% value. 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.
Here, I refers to the identity matrix, which is necessary because the distribution is multivariate. This is different to how the frequentist approach is usually outlined. In the frequentist setting above there is no mention of probability distributions for anything other than the measurement error. In the Bayesian formulation the entire problem is recast such that the yi values are samples from a normal distribution. ". What do we get out of this reformulation? There are two main reasons for doing so: • Prior Distributions: If we have any prior knowledge about the parameters β then we can choose prior distributions that reflect this.
Advanced Algorithmic Trading by Michael Halls Moore