Understanding the difference between Bayesian and frequentist approaches is fundamental to correctly interpreting credible intervals and Bayes factors. Let's break it down.

1) Understanding the Difference: Bayesian vs. Frequentist Approaches

These two statistical paradigms represent different philosophies about probability, how parameters are treated, and how inferences are drawn.

2) Interpreting Credible Intervals (CrIs)

Credible Intervals are the Bayesian equivalent of frequentist confidence intervals, but their interpretation is fundamentally different and, arguably, more intuitive.

Definition: A $X\%$ credible interval for a parameter $\theta$ is an interval within which there is an $X\%$ probability that the true value of $\theta$ lies, given the observed data and the prior distribution.

Key Differences from Confidence Intervals:

• Confidence Interval (Frequentist): If you repeat the experiment many times, $X\%$ of the confidence intervals you construct will contain the true (fixed) parameter value. You cannot say there's an $X\%$ chance the true parameter is in this specific interval. The parameter is fixed; the interval is random.


• Credible Interval (Bayesian): You can directly say there's an $X\%$ probability that the true parameter value is within this specific interval. The parameter is a random variable; the interval is fixed (given your data and prior).

• Equal-tailed interval: The most common type. If you want a 95% CrI, you find the 2.5th percentile and the 97.5th percentile of the posterior distribution. This means 2.5% of the posterior probability is in the lower tail, and 2.5% in the upper tail.


• Highest Posterior Density (HPD) interval: This is the narrowest possible credible interval for a given probability level. All points within an HPD interval have a higher posterior density than points outside the interval. For symmetric, unimodal posterior distributions (like a normal distribution), the equal-tailed and HPD intervals are the same.

Suppose you calculated a 95% credible interval for the mean treatment effect of a drug, and it is [0.5, 1.2].

• Correct: "Given the data and our prior beliefs, there is a 95% probability that the true mean treatment effect of the drug lies between 0.5 and 1.2 units."


• Also correct: "We are 95% confident that the true mean treatment effect is between 0.5 and 1.2 units." (But here, "confident" truly means probabilistic belief, not repeated sampling properties).


• Incorrect (Frequentist Misinterpretation): "If we were to repeat this study many times, 95% of the intervals we compute would contain the true mean treatment effect." (This refers to a confidence interval, not a credible interval).

3) Interpreting Bayes Factors (BFs)

Bayes Factors are used for model comparison and provide a way to quantify the evidence that the data provide for one model over another. They are an alternative to p-values for hypothesis testing.

Definition: A Bayes Factor (BF) is the ratio of the marginal likelihood of the data under one model ($M1$) to the marginal likelihood of the data under another model ($M2$).

$BF{12} = \frac{P(D|M1)}{P(D|M_2)}$

• $P(D|M_1)$ is the marginal likelihood of the data under Model 1. This means the probability of the data, averaged over all possible parameter values under Model 1, weighted by their prior probabilities.


• $P(D|M_2)$ is the marginal likelihood of the data under Model 2.

Interpretation Scale (Jeffreys' Scale, common guideline):

• Can provide evidence for the null hypothesis: If $BF_{01}$ (evidence for null over alternative) is high, it means the data are more likely under the null. P-values can only tell you if there's enough evidence to reject the null. Failing to reject the null with a p-value doesn't mean the null is true, just that you don't have enough evidence to claim otherwise.


• Not sensitive to sampling intent (stopping rules): The Bayes Factor only depends on the observed data, not on how the data were collected (e.g., if you stopped collecting data once a certain effect size was observed).


• Integrates over parameter uncertainty: It considers all possible parameter values under each model, weighted by their prior plausibility.

• Sensitivity to Priors: Bayes Factors can be quite sensitive to the choice of priors for the parameters within the models, especially for models that are not well-specified.


• Computational Complexity: Calculating marginal likelihoods can be challenging, especially for complex models, often requiring advanced simulation methods (e.g., importance sampling, bridge sampling).


• Relative Evidence: A Bayes Factor tells you which model is better relative to another, not whether either model is a good fit for the data in an absolute sense.

Suppose you are comparing a null model ($M0$: no effect) with an alternative model ($M1$: there is an effect).

• $BF_{10} = 5.2$


• Correct: "The data are 5.2 times more likely under the alternative model ($M1$) than under the null model ($M0$). This constitutes moderate evidence for the alternative hypothesis."


• Incorrect: "There is a 5.2% chance that the alternative model is true." (This is not a probability; it's a ratio of likelihoods).


• $BF{10} = 0.15$ (which is $1/BF{01} = 1/6.67$)


• Correct: "The data are 0.15 times as likely under the alternative model ($M1$) as under the null model ($M0$). Equivalently, the data are about 6.7 times more likely under the null model ($M0$) than under the alternative model ($M1$). This constitutes moderate evidence for the null hypothesis."

By understanding these distinctions and interpretations, you'll be well-equipped to perform and communicate your Bayesian analyses effectively. Remember that the choice of prior distribution is a crucial and often debated aspect of Bayesian analysis, influencing both your credible intervals and Bayes Factors.