Bayes' Theorem
Bayes' Theorem is a mathematical formula used to update the probability for a hypothesis as new evidence becomes available. It is widely used in fields such as machine learning, data science, and engineering to make predictions and decisions based on uncertain information. The theorem provides a systematic way to incorporate prior knowledge and new data to refine and improve the accuracy of predictions.
6 Key excerpts on "Bayes' Theorem"
- Michael B. Miller(Author)
- 2013(Publication Date)
- Wiley(Publisher)
...CHAPTER 6 Bayesian Analysis B ayesian analysis is an extremely broad topic. In this chapter we introduce Bayes' Theorem and other concepts related to Bayesian analysis. We will begin to see how Bayesian analysis can help us tackle some very difficult problems in risk management. OVERVIEW The foundation of Bayesian analysis is Bayes' Theorem. Bayes' Theorem is named after the eighteenth-century English mathematician Thomas Bayes, who first described the theorem. During his life, Bayes never actually publicized his eponymous theorem. Bayes' Theorem might have been confined to the dustheap of history had not a friend submitted it to the Royal Society two years after his death. Bayes' Theorem itself is incredibly simple. For two random variables, A and B, Bayes' Theorem states that: (6.1) In the next section we'll derive Bayes' Theorem and explain how to interpret Equation 6.1. As we will see, the simplicity of Bayes' Theorem is deceptive. Bayes' Theorem can be applied to a wide range of problems, and its application can often be quite complex. Bayesian analysis is used in a number of fields. It is most often associated with computer science and artificial intelligence, where it is used in everything from spam filters to machine translation and to the software that controls self-driving cars. The use of Bayesian analysis in finance and risk management has grown in recent years, and will likely continue to grow. What follows makes heavy use of joint and conditional probabilities. If you have not already done so and you are not familiar with these topics, you can review them in Chapter 2. Bayes' Theorem Assume we have two bonds, Bond A and Bond B, each with a 10% probability of defaulting over the next year. Further assume that the probability that both bonds default is 6%, and that the probability that neither bond defaults is 86%. It follows that the probability that only Bond A or Bond B defaults is 4%...
eBook - ePubOil and Gas Processing Equipment
Risk Assessment with Bayesian Networks
- G. Unnikrishnan(Author)
- 2020(Publication Date)
- CRC Press(Publisher)
...Bayes formula can be used to find out the probability of P(A|B), when we know conditional probabilities of the form P(B|A) and prior probabilities of P(A) and P(B). This way Bayes theorem can be used to describe cause and effect or causality. A typical example is a case in a process facility where there are many spurious trips. One of the causes suggested is fault of the high-pressure sensor in a particular vessel, which can be written as P(Trip|Fault of sensor). Here in this case we have a sample space that is divided into two, the event that the sensor is faulty and the event when the sensor is not faulty. Now we have the specific case of a trip and we want to hypothesize which event in the partition the specific outcome came from. We will examine the nature causality further, in terms of cause and effect, and represent the same by Bayes theorem and then discuss BN. 2.2 Bayes Theorem and Nature of Causality Bayes theorem states that if the probability of occurrence of A and B are stated as P(A) and P(B), then P(A) happening given that B has already happened can be written as P (A | B) = P(B |A) P (A) P (B) (2.10) Equation 2.10 can be rewritten as in Equation 2.11 for cause and effect, given that normally we see only the effect. P (effect | cause) = P (cause | effect) P (effect) P (cause) (2.11) It states that the probability of an effect, given that a cause has happened, can be described by a combination of the probability of cause given the. effect – which would be observable – and the unconditional probabilities of effect and cause. In the RHS of Equation 2.11, P (effect) is the prior probability, P (cause | effect) is the likelihood and P (cause) is the total probability of cause. RHS when computed will give the Left-Hand Side (LHS) known as posterior probability...
- Bruce B. Frey(Author)
- 2018(Publication Date)
- SAGE Publications, Inc(Publisher)
...Dividing through by P (B) gives a simple but powerful result that explains precisely how P (A | B) and P (B | A) are related. The next section gives several ways to express this relationship. Three Forms of Bayes’s Theorem Basic Form As explained earlier, a simple but fundamental consequence of the definition of conditional probability is the following theorem, which connects P (A | B) to P (B | A): Other Here P (A) is called the prior probability of A (it is the probability of A before we know whether B occurred), P (A | B) is the posterior probability of A given B (it is the updated probability for A, in light of the information that B occurred), and P (B) is the marginal or unconditional probability of B. Remarkably, this theorem, whose proof is essentially just one line of algebra, has deep consequences throughout. statistical theory and practice. Often P (B | A) is easier to think about or compute directly than p (A | B), or vice versa; Bayes’s theorem enables working with whichever of these is easier to handle and then bridging to the other. For example, in a criminal trial, we may be especially interested in the probability that the defendant is innocent given the evidence, but it may be easier at first to consider the probability of the evidence given that the defendant is innocent. Bayes’s theorem is named after Reverend Thomas Bayes, due to his seminal paper An Essay towards Solving a Problem in the Doctrine of Chances, which was published posthumously in 1763 with help and edits from Bayes’s friend Richard Price. Bayes’s paper established conditional probability as a powerful framework for thinking about uncertainty and derived some important properties (including Bayes’s theorem). Some historical controversies have arisen about whether anyone discovered Bayes’s theorem earlier than Bayes, and how much of a role Price played. The mathematician Pierre-Simon Laplace also played a crucial role in the early development of Bayes’s theorem...
eBook - ePub- Gary L Rosner, Purushottam W. Laud, Wesley O. Johnson(Authors)
- 2021(Publication Date)
- Chapman and Hall/CRC(Publisher)
...Chapter 2 Fundamentals I: Bayes’ Theorem, Knowledge Distributions, Prediction In this chapter we introduce the principles and basic tools of Bayesian statistical inference. In particular, we consider an approach to statistical inference that leads to making probability statements about unknown quantities of interest or events, for example the occurrence of a disease such as cancer (yes/no) in a particular patient, or the event of surviving at least 5 years after diagnosis with stage 3 breast cancer. We begin with an elementary form of Bayes’ theorem. Bayes’ theorem follows from the mathematical definition of conditional probability, which we also discuss. This elementary form of Bayes’ theorem describes how to handle unknown quantities that are dichotomous (two categories) or polychotomous (multiple categories). For example, we may sample individuals and test them for an infection, in which case there is a simple yes/no or 1/0 dichotomous outcome, or we may observe in addition, for individuals who are infected, whether they are in an early or late stage of infection, in which case the outcome is trichotomous. We provide illustrations of basic probability concepts that illustrate these probability rules before proceeding to discuss probability models for unknowns of interest, for example, the proportion of HIV infections among blood donors or the proportion of a population with uncontrolled hypertension. Next is a discussion by example of how to transform scientific knowledge about population characteristics of interest into probability models that describe and characterize that knowledge. Then we introduce and discuss the concepts of continuous and discrete random variables (RVs). Random variables are numerical outcomes associated with studies or events that are not precisely predictable. A continuous outcome in theory has a continuum of possible values, while a discrete outcome has a finite or countably infinite number of possible values...
eBook - ePubPhilosophy of Science
A Contemporary Introduction
- Alex Rosenberg, Lee McIntyre(Authors)
- 2019(Publication Date)
- Routledge(Publisher)
...Naturally, recalling the earlier success of Newton’s laws in uncovering the existence of Neptune and Uranus, the initial blame for the drop was placed on the auxiliary hypotheses. Bayes’ theorem can even show us why. Though the numbers in our example are made up, in this case, the auxiliary assumptions were eventually vindicated, and the data about the much greater than expected precession of the perihelion of Mercury undermined Newton’s theory, and (as another application of Bayes’ theorem would show), increased the probability of Einstein’s alternative theory of relativity. Philosophers and many statisticians hold that the reasoning scientists use to test their hypotheses can be reconstructed as inferences in accordance with Bayes’ theorem. These theorists are called Bayesians, and they seek to show that the history of acceptance and rejection of theories in science honors Bayes’ theorem, thus showing that in fact, theory testing has been on firm footing all along. Other philosophers and statistical theorists attempt to apply Bayes’ theorem in order to determine the probability of scientific hypotheses when the data are hard to get, sometimes unreliable, or only indirectly relevant to the hypothesis under test. For example, they seek to determine the probabilities of various hypotheses about evolutionary events, like the splitting of ancestral species from one another, by applying Bayes’ theorem to data about differences in the polynucleotide sequences of the genes of currently living species. How Much Can Bayes’ Theorem Really Help? How much understanding of the nature of empirical testing does Bayesianism really provide? Will it reconcile science’s empiricist epistemology with its commitment to unobservable events and processes that explain observable ones? Will it solve Hume’s problem of induction? To answer these questions, we must first understand what the probabilities are that all these ps symbolize and where they come from...
eBook - ePubPhilosophy of Science
A Contemporary Introduction
- Alex Rosenberg(Author)
- 2011(Publication Date)
- Routledge(Publisher)
...Other philosophers, and statistical theorists attempt to apply Bayes’ theorem actually to determine the probability of scientific hypotheses when the data are hard to get, sometimes unreliable, or only indirectly relevant to the hypothesis under test. For example, they seek to determine the probabilities of various hypotheses about evolutionary events like the splitting of ancestral species from one another, by applying Bayes’ theorem to data about differences in the polynucleotide sequences of the genes of currently living species. How Much Can Bayes’ Theorem Really Help? How much understanding of the nature of empirical testing does Bayesianism really provide? Will it reconcile science’s empiricist epistemology with its commitment to unobservable events and processes that explain observable ones? Will it solve Hume’s problem of induction? To answer these questions, we must first understand what the probabilities are that all these p’s symbolize and where they come from. We need to make sense of p(h), the probability that a certain proposition is true. There are at least two questions to be answered: First, there is the “metaphysical” question of what fact is it about the world, if any, that makes a particular probability value, p(h) for a hypothesis, h, the true or correct one? Second, there is the epistemological question of justifying our estimate of this probability value. The first question may also be understood as a question about the meaning of probability statements, and the second about how they justify inductive conclusions about general theories and future eventualities. Long before the advent of Bayesianism in the philosophy of science the meaning of probability statements was already a vexed question. There are some traditional interpretations of probability we can exclude as unsuitable interpretations for the employment of Bayes’ theorem...
