Bayesian methods are a powerful way to combine multiple sources of evidence and update your beliefs based on new data. In this article, you will learn the basics of Bayesian inference and how to apply it to different scenarios using simple examples and code.
Bayesian inference is a way of reasoning that uses probabilities to represent uncertainty and update beliefs based on evidence. The key idea is that you start with a prior probability distribution that reflects your initial assumptions or knowledge about a parameter or hypothesis, and then you update it with a likelihood function that measures how well the evidence fits your prior. The result is a posterior probability distribution that incorporates both the prior and the evidence, and gives you a range of plausible values and their probabilities for the parameter or hypothesis.
Bayes' theorem is the mathematical formula that describes how to update your prior probability with the likelihood function to get the posterior probability. It says that the posterior probability of a hypothesis given the evidence is proportional to the product of the prior probability of the hypothesis and the likelihood of the evidence given the hypothesis. In symbols, it looks like this: P(H|E) ∝ P(H) × P(E|H) where P(H|E) is the posterior probability of hypothesis H given evidence E, P(H) is the prior probability of H, P(E|H) is the likelihood of E given H, and ∝ means proportional to. To get the exact posterior probability, you need to divide by the marginal probability of E, which is the sum of the products of the prior and likelihood for all possible hypotheses. This is also called the normalizing constant or the evidence. P(H|E) = P(H) × P(E|H) / P(E) where P(E) = Σ P(H) × P(E|H) for all H.
Choosing a prior distribution is a key element of Bayesian inference, as it reflects your initial beliefs or assumptions about the parameter or hypothesis you are studying. There are various types of prior distributions, such as informative, uninformative, conjugate, and non-conjugate. When selecting a prior distribution, factors such as the domain and scale of the parameter or hypothesis, available information from previous studies or experts, the degree of uncertainty or variability you want to express, and the compatibility with the likelihood function and computational ease should all be taken into account. A common approach to choosing a prior distribution is to use a conjugate prior, which is a prior distribution that belongs to the same family as the posterior distribution when combined with a specific likelihood function. This makes calculating the posterior distribution simpler and allows you to update your prior with new evidence iteratively. For instance, if you have a binomial likelihood function for a proportion parameter, you can use a beta prior distribution, which is conjugate to the binomial likelihood and results in a beta posterior distribution.
Calculating the posterior distribution can be done in various ways, depending on the complexity of the model and data. The analytical method is the simplest and fastest, but only works for simple models with conjugate priors and likelihoods. You can use Bayes' theorem formula and the properties of the conjugate prior and likelihood to derive the posterior distribution analytically. Alternatively, numerical methods require more computation and approximation, using numerical integration or optimization techniques to find the posterior distribution or its summary statistics. Lastly, simulation methods are the most powerful and versatile, but also require the most computation and programming. You can use simulation techniques such as Monte Carlo methods or Markov chain Monte Carlo methods to generate random samples from the posterior distribution and use them to estimate its characteristics or make predictions.
Bayesian inference is a useful tool for combining multiple sources of evidence in a coherent and consistent way. Evidence can include data, prior information, expert opinions, or meta-analyses. There are various methods to combine evidence, such as hierarchical modeling, model averaging, and data fusion. Hierarchical modeling allows you to account for heterogeneity and similarity among different groups or units of analysis; model averaging deals with model uncertainty and selection bias; and data fusion integrates data from different sources or modalities. For example, hierarchical models can be used to analyze the effects of a treatment on different patients from different hospitals; model averaging can be used to forecast the outcome of an election based on different polls or methods; and data fusion can be used to measure public opinion or sentiment on a topic by combining surveys, experiments, and social media data.