The Beta(1,1) prior part is easy since it's just uniform, so the posteriors are Beta(516,9486) and Beta(501,9501).
First, recognize that with Beta(1,1) priors and binomial data, the posteriors are Beta(1+conversions, 1+non-conversions) for each arm. Then, compute the posterior distributions and estimate the probability that the difference in conversion rates (lift) is greater than zero, either analytically or via Monte Carlo simulation.
Pro tip: In practice, use Monte Carlo simulation to estimate the probability that lift > 0, as it's straightforward and scales to more complex metrics. Also, mention that with large sample sizes, the posterior is approximately normal, so you can use a normal approximation for a quick sanity check.
Note that Beta(1,1) is a uniform prior, and the data follows a binomial distribution. The posterior for each arm is Beta(α + conversions, β + non-conversions).
For arm A (515 conversions): posterior is Beta(1+515, 1+9485) = Beta(516, 9486). For arm B (500 conversions): posterior is Beta(1+500, 1+9500) = Beta(501, 9501).
Use Monte Carlo simulation: draw many samples from each posterior, compute the difference (or ratio) of conversion rates, and calculate the proportion of samples where the difference is positive.
Report the probability that lift > 0, and optionally provide a credible interval for the lift. Discuss practical significance and any assumptions.
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Start by defining expected loss as the weighted sum of false positive and false negative costs, then derive the optimal decision threshold that minimizes this loss. Explain how the threshold shifts based on the cost ratio, and discuss practical implications for model deployment and evaluation.
Pro tip: Emphasize that in real-world applications, the cost ratio is often estimated from business context, and you should validate the chosen threshold using a holdout set or cross-validation to ensure robustness.
Clearly state the costs associated with false positives (C_FP) and false negatives (C_FN), and note that these are asymmetric. Explain that these costs can be monetary, time, or other business metrics.
Express the expected loss for a given decision threshold as: E[Loss] = C_FP * P(FP) + C_FN * P(FN), where P(FP) and P(FN) depend on the threshold and the model's score distribution.
Show that the optimal threshold is where the likelihood ratio equals the cost ratio: P(x|positive)/P(x|negative) = C_FP / C_FN. Alternatively, set the threshold such that the posterior probability of positive class equals C_FP / (C_FP + C_FN).
Explain how the threshold shifts: if false negatives are more costly, lower the threshold to predict positive more often. Discuss how to estimate costs and validate the threshold on validation data.
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Optional stopping is where frequentist tests fall apart and I said so pretty directly.
Start by contrasting the philosophical and practical implications of Bayesian and frequentist approaches under optional stopping, emphasizing that Bayesian inference remains valid while frequentist p-values become miscalibrated. Then, discuss how to translate posterior probabilities into actionable insights for stakeholders, avoiding technical jargon and focusing on decision-making.
Pro tip: Frame the Bayesian approach as a tool for continuous monitoring and decision-making, which aligns with business needs, while cautioning that frequentist methods require pre-specified stopping rules to avoid inflated error rates. This shows you understand both the statistical rigor and the practical trade-offs.
Explain that Bayesian inference updates beliefs continuously and is not affected by optional stopping, whereas frequentist p-values assume a fixed sample size and become invalid if you peek at data repeatedly.
Describe how optional stopping leads to inflated Type I error in frequentist tests, while Bayesian posterior probabilities remain coherent. Mention that Bayesian methods allow for more flexible experimentation but require prior specification.
For posterior probabilities, use plain language like 'There is an 85% chance that variant B is better' and tie it to expected loss or gain. For p-values, explain them as 'the probability of seeing this result if there were no effect' and highlight their limitations under peeking.
Emphasize that Bayesian posterior probabilities directly answer the question stakeholders care about: 'What is the probability that B is better?' while p-values answer a different, less intuitive question.
Suggest that for Meta's scale, Bayesian methods can enable faster, more intuitive decisions, but acknowledge that frequentist methods are simpler and more familiar. Recommend a hybrid or clear guidelines depending on the context.
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Start by acknowledging that over-optimism often stems from biased prior elicitation, such as relying on stakeholder opinions or historical data without adjustment. Then, describe a structured process that combines multiple sources of evidence, uses conservative adjustments, and validates priors through calibration. Emphasize the importance of pre-registration and sensitivity analysis to detect and mitigate over-optimism.
Pro tip: Frame priors as a range rather than a point estimate, and explicitly discuss how you would detect and correct for over-optimism if early experiment results deviate from prior expectations.
Gather priors from multiple independent sources: historical experiments, domain expertise, and analogous studies. Avoid relying solely on optimistic stakeholder projections.
Systematically shrink overly optimistic priors toward more neutral values using techniques like empirical Bayes or discounting factors based on past miscalibration.
Document priors before the experiment and conduct sensitivity analyses to see how results change under different prior assumptions, including pessimistic scenarios.
After the experiment, compare prior predictions with observed outcomes to assess calibration. Use this feedback to refine future prior elicitation processes.
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