Bayesian Statistics

Extracted Attributes

• Field

  Statistics and Probability Theory

• Limitations

  Historically high computational demands and philosophical differences with frequentist approaches

• Core Formula

  Bayes' Theorem

• Key Components

  Prior distribution, Likelihood function, and Posterior distribution

• Modern Application

  Real-time data analysis for drug approval and clinical trials

• Computational Methods

  Markov chain Monte Carlo (MCMC) algorithms

• Primary Interpretation

  Degree of belief or confidence in an event

Timeline

• Birth of Thomas Bayes, the English statistician for whom the theory is named. (Source: IBM)

  1701-01-01

• Posthumous publication of Thomas Bayes' paper formulating a specific case of Bayes' theorem. (Source: Wikipedia)

  1763-12-31

• Approximate period during which Pierre-Simon Laplace developed the Bayesian interpretation of probability in several papers. (Source: Wikipedia)

  1812-01-01

• The term 'Bayesian' becomes commonly used to describe these statistical methods. (Source: Wikipedia)

  1950-01-01

• Bayesian methods gain significant prominence in the 21st century due to increased computing power and MCMC algorithms. (Source: Wikipedia)

  2001-01-01

• FDA Commissioner Marty Makary proposes adopting Bayesian statistics for real-time data analysis during the JP Morgan Healthcare Conference in San Francisco, USA. (Source: Document 065d2e96-4d40-49bd-8511-d8d35f8b01f4)

  2025-01-01

Bayesian statistics

Bayesian statistics ( BAY-zee-ən or BAY-zhən) is a theory in the field of statistics based on the Bayesian interpretation of probability, where probability expresses a degree of belief in an event. The degree of belief may be based on prior knowledge about the event, such as the results of previous experiments, or on personal beliefs about the event. This differs from a number of other interpretations of probability, such as the frequentist interpretation, which views probability as the limit of the relative frequency of an event after many trials. More concretely, analysis in Bayesian methods codifies prior knowledge in the form of a prior distribution. Bayesian statistical methods use Bayes' theorem to compute and update probabilities after obtaining new data. Bayes' theorem describes the conditional probability of an event based on data as well as prior information or beliefs about the event or conditions related to the event. For example, in Bayesian inference, Bayes' theorem can be used to estimate the parameters of a probability distribution or statistical model. Since Bayesian statistics treats probability as a degree of belief, Bayes' theorem can directly assign a probability distribution that quantifies the belief to the parameter or set of parameters. Bayesian statistics is named after Thomas Bayes, who formulated a specific case of Bayes' theorem in a paper published in 1763. In several papers spanning from the late 18th to the early 19th centuries, Pierre-Simon Laplace developed the Bayesian interpretation of probability. Laplace used methods now considered Bayesian to solve a number of statistical problems. While many Bayesian methods were developed by later authors, the term "Bayesian" was not commonly used to describe these methods until the 1950s. Throughout much of the 20th century, Bayesian methods were viewed unfavorably by many statisticians due to philosophical and practical considerations. Many of these methods required much computation, and most widely used approaches during that time were based on the frequentist interpretation. However, with the advent of powerful computers and new algorithms like Markov chain Monte Carlo, Bayesian methods have gained increasing prominence in statistics in the 21st century.

Web Search Results

• Bayesian statistics - Wikipedia

  Bayesian statistics (/ˈbeɪziən/ BAY-zee-ən or /ˈbeɪʒən/ BAY-zhən) is a theory in the field of statistics based on the Bayesian interpretation of probability, where probability expresses a degree of belief in an event "Event (probability theory)"). The degree of belief may be based on prior knowledge about the event, such as the results of previous experiments, or on personal beliefs about the event. This differs from a number of other interpretations of probability, such as the frequentist interpretation, which views probability as the limit of the relative frequency of an event after many trials. More concretely, analysis in Bayesian methods codifies prior knowledge in the form of a prior distribution. [...] Bayesian statistics is named after Thomas Bayes, who formulated a specific case of Bayes' theorem in a paper published in 1763. In several papers spanning from the late 18th to the early 19th centuries, Pierre-Simon Laplace developed the Bayesian interpretation of probability. Laplace used methods now considered Bayesian to solve a number of statistical problems. While many Bayesian methods were developed by later authors, the term "Bayesian" was not commonly used to describe these methods until the 1950s. Throughout much of the 20th century, Bayesian methods were viewed unfavorably by many statisticians due to philosophical and practical considerations. Many of these methods required much computation, and most widely used approaches during that time were based on the frequentist [...] Bayesian statistical methods use Bayes' theorem to compute and update probabilities after obtaining new data. Bayes' theorem describes the conditional probability of an event based on data as well as prior information or beliefs about the event or conditions related to the event. For example, in Bayesian inference, Bayes' theorem can be used to estimate the parameters of a probability distribution or statistical model. Since Bayesian statistics treats probability as a degree of belief, Bayes' theorem can directly assign a probability distribution that quantifies the belief to the parameter or set of parameters.

• What are Bayesian statistics? | IBM

  # What is Bayesian statistics? ## Author Joshua Noble Data Scientist ## What is Bayesian statistics? Bayesian statistics is an approach to statistical inference grounded in Bayes’ theorem to update the probability of a hypothesis as more evidence or data becomes available. That theorem gives a formal definition for how prior beliefs about uncertain quantities should be updated with newly observed data to produce an estimate of the likelihood of an event happening. There are two fundamental paradigms for statistical inference: Bayesian statistics, which treats of unknown parameters parameters as random variables characterized by probability distributions, and Frequentist statistics, which treats all unknown parametersparameters as unknown constants. [...] In Bayesian statistics, parameters, predictions and even hypotheses are treated as random variables with probability distributions. Rather than producing a single estimate (e.g., a maximum likelihood point), the Bayesian framework gives a full distribution, the posterior probabilities of all features of the model, that reflects the entire range of plausible values and their relative credibility.2 This means uncertainty isn’t something added on after the fact (like a standard error), but is intrinsic to the inference process. [...] Bayesian statistics are named for the Reverend Thomas Bayes, an English statistician born in 1701. Bayes became interested in the problem of inverse probability in 1755 and formulated what became known as Bayes’ Theorem (or Bayes’ Rule) sometime between 1755 and his death in 1761. Bayes was exploring techniques to compute a distribution for the probability parameter of a binomial distribution across multiple identical Bernoulli trials. His theorem computes the reverse conditional probability of an event, and it states: - represents a hypothesis - is the observed data

• Bayesian Statistics: A Beginner's Guide - QuantStart

  In the article we will: ## What is Bayesian Statistics? Bayesian statistics is a particular approach to applying probability to statistical problems. It provides us with mathematical tools to update our beliefs about random events in light of seeing new data or evidence about those events. In particular Bayesian inference interprets probability as a measure of believability or confidence that an individual may possess about the occurance of a particular event. We may have a prior belief about an event, but our beliefs are likely to change when new evidence is brought to light. Bayesian statistics gives us a solid mathematical means of incorporating our prior beliefs, and evidence, to produce new posterior beliefs. [...] Bayesian statistics provides us with mathematical tools to rationally update our subjective beliefs in light of new data or evidence. This is in contrast to another form of statistical inference, known as classical or frequentist statistics, which assumes that probabilities are the frequency of particular random events occuring in a long run of repeated trials. For example, as we roll a fair (i.e. unweighted) six-sided die repeatedly, we would see that each number on the die tends to come up 1/6 of the time. Frequentist statistics assumes that probabilities are the long-run frequency of random events in repeated trials. [...] In order to begin discussing the modern techniques, we must first gain a solid understanding in the underlying mathematics and statistics that underpins these models. One of the key modern areas is that of Bayesian Statistics. We have not yet discussed Bayesian methods in any great detail on the site. This article has been written to help you understand the "philosophy" of the Bayesian approach, how it compares to the traditional/classical frequentist approach to statistics and the potential applications in both quantitative finance and data science. In the article we will: ## What is Bayesian Statistics?

• Bayesian statistics and modelling | Nature Reviews Methods Primers

  Bayesian statistics is an approach to data analysis based on Bayes’ theorem, where available knowledge about parameters in a statistical model is updated with the information in observed data. The background knowledge is expressed as a prior distribution and combined with observational data in the form of a likelihood function to determine the posterior distribution. The posterior can also be used for making predictions about future events. This Primer describes the stages involved in Bayesian analysis, from specifying the prior and data models to deriving inference, model checking and refinement. We discuss the importance of prior and posterior predictive checking, selecting a proper technique for sampling from a posterior distribution, variational inference and variable selection.

• Chapter 1 The Basics of Bayesian Statistics

  # Chapter 1 The Basics of Bayesian Statistics Bayesian statistics mostly involves conditional probability, which is the the probability of an event A given event B, and it can be calculated using the Bayes rule. The concept of conditional probability is widely used in medical testing, in which false positives and false negatives may occur. A false positive can be defined as a positive outcome on a medical test when the patient does not actually have the disease they are being tested for. In other words, it’s the probability of testing positive given no disease. Similarly, a false negative can be defined as a negative outcome on a medical test when the patient does have the disease. In other words, testing negative given disease. Both indicators are critical for any medical decisions. [...] | Model (p) | 0.1000 | 0.2000 | 0.3000 | 0.4000 | 0.5000 | 0.6000 | 0.70 | 0.80 | 0.90 | | Prior P(model) | 0.0600 | 0.0600 | 0.0600 | 0.0600 | 0.5200 | 0.0600 | 0.06 | 0.06 | 0.06 | | Likelihood P(data|model) | 0.0898 | 0.2182 | 0.1304 | 0.0350 | 0.0046 | 0.0003 | 0.00 | 0.00 | 0.00 | | P(data|model) x P(model) | 0.0054 | 0.0131 | 0.0078 | 0.0021 | 0.0024 | 0.0000 | 0.00 | 0.00 | 0.00 | | Posterior P(model|data) | 0.1748 | 0.4248 | 0.2539 | 0.0681 | 0.0780 | 0.0005 | 0.00 | 0.00 | 0.00 | [...] We started with the high prior at \(p=0.5\), but the data likelihood peaks at \(p=0.2\). And we updated our prior based on observed data to find the posterior. The Bayesian paradigm, unlike the frequentist approach, allows us to make direct probability statements about our models. For example, we can calculate the probability that RU-486, the treatment, is more effective than the control as the sum of the posteriors of the models where \(p<0.5\). Adding up the relevant posterior probabilities in Table 1.2, we get the chance that the treatment is more effective than the control is 92.16%. ### 1.2.3 Effect of Sample Size on the Posterior The RU-486 example is summarized in Figure 1.1, and let’s look at what the posterior distribution would look like if we had more data.