advanced-algorithmic-trading

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3.4.3 Multiple Flips of the Coin
Now that we have the Bernoulli likelihood function we can use it to determine the probability of
seeing a particular sequence of N flips, given by the set {k 1 , ..., k N }.
Since each of these flips is independent of any other, the probability of the sequence occuring
is simply the product of the probability of each flip occuring.
If we have a particular fairness parameter θ, then the probability of seeing this particular
stream of flips, given θ, is as follows:
P ({k 1 , ..., k N }|θ) = ∏ i
= ∏ i
P (k i |θ) (3.7)
θ ki (1 − θ) 1−ki (3.8)
What if we are interested in the number of heads, say, in N flips? If we denote by z the
number of heads appearing, then the formula above becomes:
P (z, N|θ) = θ z (1 − θ) N−z (3.9)
That is, the probability of seeing z heads in N flips assuming a fairness parameter θ. We
will use this formula when we come to determine our posterior belief distribution later in the
chapter.
3.5 Quantifying our Prior Beliefs
An extremely important step in the Bayesian approach is to determine our prior beliefs and then
find a means of quantifying them.
In the Bayesian approach we need to determine our prior beliefs on parameters and
then find a probability distribution that quantifies these beliefs.
In this instance we are interested in our prior beliefs on the fairness of the coin. That is, we
wish to quantify our uncertainty in how biased the coin is.
To do this we need to understand the range of values that θ can take and how likely we think
each of those values are to occur.
θ = 0 indicates a coin that always comes up tails, while θ = 1 implies a coin that always
comes up heads. A fair coin is denoted by θ = 0.5. Hence θ ∈ [0, 1]. This implies that our
probability distribution must also exist on the interval [0, 1].
The task then becomes determining which probability distribution we utilise to quantify our
beliefs about the coin.
3.5.1 Beta Distribution
In this instance we are going to choose the beta distribution. The probability density function
(PDF) of the beta distribution is given by the following: