Theory

The Frequentist approach, also known as classical statistics, is based on the idea that probabilities represent the long-run frequency of events occurring. The key assumption in Frequentist statistics is that the true parameter values are fixed and unknown, while the data is considered random.

Mathematically, Frequentists rely on concepts like confidence intervals and hypothesis testing to capture uncertainty. For a confidence interval, we estimate a range of values within which the true population parameter is expected to lie, with a certain level of confidence. For a sample mean, \(\bar{X}\), the confidence interval can be calculated as:

\[ \bar{X} \pm z \frac{\sigma}{\sqrt{n}} \]

Where \(z\) is the critical value from the standard normal distribution, \(\sigma\) is the population standard deviation, and \(n\) is the sample size.

Philosophy

The Frequentist approach emphasizes objective probabilities derived from the relative frequencies of events observed in the past. It considers the true parameter values as fixed and strives to estimate them using data, while acknowledging that the data itself is subject to randomness.

Example

If a scientist wants to estimate the average height of a certain species of plant, they would collect a random sample, calculate the sample mean and standard deviation, and construct a confidence interval. The width of the confidence interval reflects the level of uncertainty – a wider interval suggests greater uncertainty, while a narrower interval implies more certainty about the population parameter.

Theory

The Bayesian approach treats probabilities as a subjective degree of belief rather than an objective long-run frequency. In this approach, uncertainty is quantified through the use of probability distributions that represent our prior beliefs about the parameters, which are then updated with observed data to obtain posterior distributions.

Bayesian methods rely on the concept of conditional probability and Bayes’ theorem to capture uncertainty mathematically. Bayes’ theorem states: \[ P(H \mid D)=\frac{P(D \mid H) P(H)}{P(D)} \]

Where \(P(H|D)\) is the posterior probability of hypothesis \(H\) given data \(D\), \(P(D|H)\) is the likelihood of data \(D\) given hypothesis \(H\), \(P(H)\) is the prior probability of hypothesis \(H\), and \(P(D)\) is the probability of data \(D\).

Philosophy

The Bayesian approach emphasizes subjective belief and parameter updating. It treats both data and parameters as random variables, and incorporates prior knowledge and evidence to refine our beliefs about the parameters.

Example

For instance, if a doctor wants to determine the probability of a patient having a particular disease based on a test result, they would start with a prior probability (based on the prevalence of the disease in the population) and update it with the likelihood of the test result, given the disease status. The resulting posterior probability provides the doctor with a more refined estimate of the patient’s probability of having the disease, reflecting the uncertainty inherent in the diagnostic process.