Introduction

Bayesian statistics is a method of statistical inference named after the English mathematician Thomas Bayes. Its main purpose is to combine prior information about a population parameter with data derived from a sample in order to guide the process of statistical inference. Unlike classical statistics, it does not rely exclusively on observed data but also incorporates prior knowledge, thus offering a more complete picture of uncertainty.

Steps of Bayesian Analysis

Bayesian analysis can be described through four main steps. The first step is to determine the data available. For example, if we consider the toss of a coin with two possible outcomes, heads or tails, we can use the random variable Y, which takes the value 1 when heads occurs and 0 when tails occurs. In this way, the data are given a simple and mathematically manageable form.

The second step is the creation of a probabilistic model that represents the data-generating process. In the case of the coin toss, the probability of obtaining heads is denoted by the parameter θ. This parameter defines the binomial distribution that constitutes our model and allows us to describe the behavior of the data in a formal way.

The third step is the specification of prior distributions. At this stage, we define which values of the parameter θ are considered more plausible before observing the data. For instance, it is reasonable to assume that a coin is more likely to be fair, so θ equal to 0.5 would have higher probability than other values. Thus, the prior distribution expresses our initial beliefs.

The fourth and final step is the application of Bayes’ rule. At this point, we use the observed data to update our prior beliefs and obtain the posterior distribution. For example, if the first toss of the coin results in heads, we can compute the probability of each value of θ given this new observation. The posterior distribution therefore represents the compromise between prior knowledge and the information provided by the sample.

Bayes Factor and Hypothesis Testing

Bayesian analysis places particular emphasis on hypothesis testing. The Bayes factor is used to compare the null hypothesis with an alternative hypothesis. In practice, this means examining whether the data support H0 or the alternative H1 more strongly. The testing process is a matter of decision-making, as we must reject or accept the null hypothesis. As in classical statistics, there are two types of errors: Type I error, when H0 is rejected while it is true, and Type II error, when H0 is accepted while H1 is true. Bayesian statistics employs loss functions to provide a mathematical framework for the consequences of such decisions, often assuming that both types of error are equally important.

Test Functions and Decision

In classical statistics, hypothesis testing is often carried out through test functions, which define the critical region for rejecting the null hypothesis. The Bayesian method follows a similar logic, as the posterior distribution determines the region of acceptance or rejection of H0. The decision-making process is therefore based on the posterior probability resulting from the combination of prior distributions and observed data, making the final decision more closely adapted to reality.

Bayesian Structural Time Series

A particularly interesting application of the Bayesian approach is structural time series. Their general model includes a trend, a seasonal component, and a noise term. Within the framework of Bayesian analysis, the components are estimated using algorithms such as the Kalman filter and the Kalman smoother, while the Spike and Slab method is often used for coefficient estimation. In this way, the appropriate model can be fitted to the data, and more accurate forecasts can be made. The advantage of the Bayesian approach is that it incorporates prior information about parameters and accounts for uncertainty in predictions, making them more realistic in settings where data are limited or noisy.

Conclusion

Bayesian analysis is one of the most important and flexible methods of modern statistics. Through the steps of data specification, model creation, prior distribution determination, and application of Bayes’ rule, it achieves a comprehensive estimation that combines theory with empirical evidence. The Bayes factor and test functions provide rigor in hypothesis testing, while Bayesian structural time series highlight the practical utility of the method in forecasting. Its main value lies in transforming uncertainty into measurable probabilities, offering a powerful tool for scientific analysis and decision-making.