Classical (or frequentist) and Bayesian statistics are the two main branches of statistical inference, but they have fundamentally different interpretations of what probabilities represent and how they should be used. This leads to differences in the application and interpretation of their methods.
1. Interpretation of Probability:
Classical Statistics: In the frequentist viewpoint, probabilities represent long-term frequencies of events. For example, if a fair coin is tossed repeatedly, we expect it to land on heads about 50% of the time. Here, probabilities are objective properties of the real world and are not associated with any degree of belief or knowledge about an event.
Bayesian Statistics: In contrast, Bayesian statistics interprets probabilities as degrees of belief. For example, if a Bayesian says there is a 50% probability of rain tomorrow, it means they believe that the chance of rain is equivalent to the chance of a fair coin landing on heads - it's a subjective belief about a single event, not a long-term frequency.
2. Use of Prior Information:
Classical Statistics: Classical methods do not incorporate prior information about unknown parameters. They use only the data at hand to make inferences about the population. This can be beneficial because it avoids the potential subjectivity introduced by the choice of prior.
Bayesian Statistics: Bayesian methods, on the other hand, combine prior information with the data to form a posterior distribution. This allows for the integration of prior knowledge or beliefs into the analysis, which can be particularly helpful in cases of limited data.
3. Hypothesis Testing and Confidence Intervals:
Classical Statistics: Classical statistics typically uses null hypothesis significance testing (NHST) and p-values for hypothesis testing. In this framework, we calculate the probability of obtaining the observed data (or more extreme data) given that a null hypothesis is true. Additionally, a 95% confidence interval means that if we were to repeat the experiment many times, 95% of the calculated intervals would contain the true population parameter.
Bayesian Statistics: Bayesian methods provide direct probabilities about hypotheses or parameters via the posterior distribution. For example, a 95% Bayesian credible interval can be interpreted as the interval within which the parameter lies with 95% probability. This is often considered more intuitive than the frequentist interpretation of confidence intervals.
4. Computational Complexity:
Classical Statistics: Classical methods often involve simpler calculations and can be less computationally intensive than Bayesian methods.
Bayesian Statistics: Bayesian methods, especially for complex models, often require sophisticated computational techniques like Markov Chain Monte Carlo (MCMC). However, the advent of powerful computers and algorithms has made these computations increasingly feasible.
Both frameworks have their strengths and weaknesses and are useful in different contexts. The choice between classical and Bayesian statistics should be based on the specific requirements of the analysis, including the nature of the problem, the available data, and the practical implications of the results.
Practical Example of Classical and Bayesian Statistics
Assume we have data for the past 50 years and we are interested in predicting whether or not it will rain tomorrow.
Classical Approach:
Let's say we are interested in the proportion of rainy days. We would calculate the sample proportion of rainy days (let's say it's 0.3 or 30% in our dataset) and create a confidence interval for this proportion. We might use a method like bootstrapping or a formula-based method (like the one for a binomial proportion) to construct this interval.
Suppose our 95% confidence interval for the population proportion of rainy days is (0.25, 0.35). In the classical framework, we would interpret this as: "If we were to collect many samples and construct a confidence interval from each sample, about 95% of these intervals would contain the true proportion of rainy days."
Note that in this classical framework, we do not incorporate any prior beliefs we might have about the proportion of rainy days, and we do not make a probabilistic statement about tomorrow's weather.
Bayesian Approach:
In the Bayesian framework, we would first specify a prior distribution that represents our beliefs about the proportion of rainy days before seeing the data. Suppose we believe that all proportions are equally likely, so we specify a uniform prior distribution.
After observing the data, we update our beliefs to obtain a posterior distribution for the proportion of rainy days. Suppose our posterior distribution is a Beta distribution (which is the conjugate prior for a binomial likelihood) with parameters that were updated based on the data.
We could then use this posterior distribution to make a probabilistic statement about tomorrow's weather. For example, we could find the probability that tomorrow is a rainy day as the posterior mean (let's say it's 0.31 or 31%). We could also create a 95% credible interval, let's say it's (0.26, 0.36). We would interpret this as "given the data, we believe that the true proportion of rainy days is between 26% and 36% with a 95% probability".
Notice that in this Bayesian framework, we incorporated our prior beliefs (albeit vague in this case), updated these beliefs based on the data, and made a probabilistic statement about a future event.
Comparison:
While both methods gave us similar estimates and intervals, the interpretations are quite different. The classical method gave us a range of values that would contain the true proportion in a long series of repetitions of the same sampling procedure, while the Bayesian method gave us a range of values that we believe contains the true proportion with a certain probability.
Moreover, the Bayesian method allowed us to make a probabilistic statement about a future event (tomorrow's weather), which was not straightforward in the classical framework.
Thus, while the computations might be similar, the philosophical differences between classical and Bayesian statistics led to different interpretations and different types of conclusions.