Bayes' factor

(noun)

The ratio of the conditional probabilities of the event $B$ given that $A_1$ is the case or that $A_2$ is the case, respectively.

Examples of Bayes' factor in the following topics:

  • Bayes' Rule

    • In probability theory and statistics, Bayes' theorem (or Bayes' rule ) is a result that is of importance in the mathematical manipulation of conditional probabilities.
    • The relationship is expressed in terms of the likelihood ratio, or Bayes' factor.
    • More specifically, given events $A_1$, $A_2$ and $B$, Bayes' rule states that the conditional odds of $A_1:A_2$ given $B$ are equal to the marginal odds $A_1:A_2$ multiplied by the Bayes factor or likelihood ratio.
    • Rationally, Bayes' rule makes a great deal of sense.
    • Furthermore, Bayes' rule can be applied iteratively.

  • Bayes' Theorem

    • In these cases, we can apply a very useful and general formula: Bayes' Theorem.
    • Bayes' Theorem states that this conditional probability can be identified as the following fraction:
    • Bayes' Theorem is just a generalization of what we have done using tree diagrams.
    • Use Bayes' Theorem only when there are so many scenarios that drawing a tree diagram would be complex.
    • Here we solve the same problem presented in Exercise 2.57, except this time we use Bayes' Theorem.

  • Conditional Probability

    • In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule) is a result that is of importance in the mathematical manipulation of conditional probabilities.
    • Mathematically, Bayes' theorem gives the relationship between the probabilities of $A$ and $B$, $P(A)$ and $P(B)$, and the conditional probabilities of $A$ given $B$ and $B$ given $A$.

  • Base Rates

    • Compute the probability of a condition from hits, false alarms, and base rates using Bayes' Theorem
    • This same result can be obtained using Bayes' theorem.
    • Bayes' theorem considers both the prior probability of an event and the diagnostic value of a test to determine the posterior probability of the event.
    • Bayes' theorem shown below allows you to calculate P(D|T), the probability that you have the disease given that you test positive for it.

  • Ethiopia and Eritrea

    • Bitter religious conflicts contributed to hostility toward foreign Christians and Europeans, which persisted into the 20th century and were a factor in Ethiopia's isolation until the mid-19th century, when the first British mission, sent in 1805 to conclude an alliance with Ethiopia and obtain a port on the Red Sea in case France conquered Egypt.
    • The Royal Enclosure (Fasil Ghebbi) and Gondar, photo by Baye Amsalo, source: Wikipedia.

  • Glossary

    • Bayes' theorem considers both the prior probability of an event and the diagnostic value of a test to determine the posterior probability of the event.
    • For example, if the factor were drug dosage, and three doses were tested, then each dosage would be one level of the factor and the factor would have three levels.
    • In a design with two factors, the marginal means for one factor are the means for that factor averaged across all levels of the other factor.
    • Also called a "repeated measures factor. "
    • The simple effect of a factor is the effect of that factor at a single level of another factor.

  • Factors

    • This is a complete list of the factors of 24.
    • Therefore, 2 and 3 are prime factors of 6.
    • However, 6 is not a prime factor.
    • To factor larger numbers, it can be helpful to draw a factor tree.
    • This factor tree shows the factorization of 864.

  • Introduction to Factoring Polynomials

    • Factoring by grouping divides the terms in a polynomial into groups, which can be factored using the greatest common factor.
    • Factor out the greatest common factor, $4x(x+5) + 3y(x+5)$.
    • Factor out binomial $(x+5)(4x+3y)$.
    • One way to factor polynomials is factoring by grouping.
    • Both groups share the same factor $(x+5)$, so the polynomial is factored as

  • Regulation of Sigma Factor Activity

    • Sigma factors are proteins that function in transcription initiation .
    • The activity of sigma factors within a cell is controlled in numerous ways.
    • However, if transcription of genes is not required, sigma factors will not be active.
    • The anti-sigma factors will bind to the RNA polymerase and prevent its binding to sigma factors present at the promoter site.
    • The anti-sigma factors are responsible for regulating inhibition of transcriptional activity in organisms that require sigma factor for proper transcription initiation.

  • French Explorers

    • By the first decades of the 18th century, the French created and controlled such colonies as Quebec, La Baye des Puants (present-day Green Bay), Ville-Marie (Montreal), Fort Pontchartrain du Détroit (modern-day Detroit), or La Nouvelle Orléans (New Orleans) and Baton Rouge.