Bayes factor

Builds on Likelihood function Prior probability Bayes' theorem

Definition

Let $M_0$ and $M_1$ be two models for data $x$ . For $i = 0, 1$ , model $M_i$ has a parameter $\theta_i \in \Theta_i$ , a density or mass function $f_i(x \mid \theta_i)$ and a proper prior $\pi_i$ on $\Theta_i$ . The marginal likelihood of $M_i$ is $m_i(x) = \int_{\Theta_i} f_i(x \mid \theta_i) \, \pi_i(d\theta_i).$ If $m_0(x) > 0$ , the Bayes factor of $M_1$ against $M_0$ is $B_{10}(x) = \frac{m_1(x)}{m_0(x)}.$

Remarks

By Bayes' theorem, with prior model probabilities $P(M_0), P(M_1) > 0$ , the posterior odds are $\frac{P(M_1 \mid x)}{P(M_0 \mid x)} = B_{10}(x) \cdot \frac{P(M_1)}{P(M_0)}.$ If both models are simple, that is each $\Theta_i$ is a single point, then $B_{10}$ is the likelihood ratio. Example: let $M_0$ say that a coin is fair and $M_1$ that its chance $p$ of heads is uniform on $[0, 1]$ , and observe two heads in two tosses. Then $m_0 = 1/4$ and $m_1 = \int_0^1 p^2 \, dp = 1/3$ , so $B_{10} = 4/3$ . Non-example: with an improper prior $\pi_1$ , the integral $m_1(x)$ is defined only up to an arbitrary constant factor, so the ratio is not a well-defined Bayes factor.

Reading path

To understand this page from the ground up, read these 66 pages in order. The foundations come first.

Construction of the reals from Cauchy sequences concept
Element definition
Integers object
Ordered field definition
Proposition concept
Random experiment concept
Rational numbers object
Real numbers object
Set definition
Ring of integers statement
Conjunction definition
Disjunction definition
Implication definition
Predicate definition
Empty set object
Family of sets definition
Ordered pair definition
Intersection definition
Union definition
Sample space definition
The rationals form a field statement
Existential quantifier definition
Universal quantifier definition
Disjoint sets definition
Cartesian product definition
Subset definition
Union of a family of sets definition
The rationals form an ordered field statement
Uniqueness quantifier definition
Universal set definition
Binary relation definition
Partition of a set definition
Order on the reals definition
Power set definition
Complement definition
Function definition
Interval definition
Injective function definition
Preimage of a set definition
Sigma-algebra definition
Countable set definition
Event definition
Borel sigma-algebra definition
Probability axioms concept
Probability measure definition
Probability distribution definition
Probability space definition
Conditional probability definition
Random variable definition
Cumulative distribution function definition
Discrete random variable definition
Law of a random variable definition
Independence of two events definition
Law of total probability statement
Random variable with a density definition
Probability mass function definition
Mutual independence of events definition
Bayes' theorem statement
Identically distributed random variables definition
Probability density function definition
Independent sigma-algebras definition
Prior probability concept
Independence of random variables definition
Independent and identically distributed random variables definition
Likelihood function definition
Bayes factor definition

Where this leads

1 page builds on this one.

Lindley's paradox

Tag	`5ZEGZVZBTJ`
URI	`m2://5ZEGZVZBTJ`
Kind	definition
Writing	W0 stub
MSC	62F15, 62F03
Defines	Bayes factor
Uses	Likelihood function, Prior probability, Bayes' theorem

Bayes factor

Definition

Remarks

Reading path

Where this leads

Further reading

What this page needs for W1

What links here

Used by