Update beta distribution parameters

[MustaphaU]

Issue #, if available:

Description of changes:

Update the beta distribution parameters in the simulate_experiment function to avoid bias towards lower success probability.

The current specification of the beta distribution:

theta = np.random.beta(conversions + 1, exposures + 1)

treats every exposure as a failure, that is overstates the failures thus undervalues the success probabilities of the variations. The effect is pronounced for variations with very high baseline conversion rates but less severe for variations with extremely low conversion rates.

Traditionally, the Thompson Sampling Algorithm for the Bernoulli Bandit Thompson Sampling algorithm is:

$$\begin{align*} 1: & \text{for } t = 1, 2, \ldots \text{ do:} \\\ 2: & \quad \quad \text{Sample model:} \\\ 3: & \quad \quad \text{for } k = 1 \text{ to } K \text{ do:} \\\ 4: & \quad \quad \quad \text{Sample } \theta_k \sim \text{beta}(\alpha_k, \beta_k) \\\ 5: & \quad \quad \text{end for} \\\ 6: & \quad \quad \text{Select and apply action:} \\\ 7: & \quad \quad x_t \leftarrow \arg\max_k \theta_k \\\ 8: & \quad \quad \text{Apply } x_t \text{ and observe } r_t \\\ 9: & \quad \quad \text{Update distribution:} \\\ 10: & \quad \quad (\alpha_{x_t}, \beta_{x_t}) \leftarrow (\alpha_{x_t} + r_t, \beta_{x_t} + 1 - r_t) \\\ 11: & \text{end for} \end{align*}$$

Where α, β represent the parameters of each arm i.e. the success and failure counts, respectively OR the number of conversions and non-conversions, respectively.

non-conversions (or beta)  = exposures - conversions

Description of testing performed to validate your changes (required if pull request includes CloudFormation or source code changes):

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[james-jory]

Thank you for catching this oversight!

[MustaphaU]

Thank you for catching this oversight!

Thank you @james-jory !
#647