Ratio of uniforms and beyond
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The document discusses the concept of the ratio of uniforms, detailing the mathematical framework for generating distributions proportional to a given function f using transformations and change of variables. It explains the process of simulating a uniform distribution and identifies applications to standard distributions, such as Student's t. The principles are generalized for monotone transformations and explore conditions for density adequacy.
![Ratio of uniforms: what?
Consider the set A of (u, v)’s in R+ × X such that
0 u2
f (v/u)
Then a uniform distribution on A induces the distribution with
density proportional to f on V /U.
[Kinderman and Monahan’s (1977)]](https://image.slidesharecdn.com/ratiou-161024122138/85/Ratio-of-uniforms-and-beyond-2-320.jpg)
![Ratio of uniforms: why?
There exists a compact box B containing A iff
0 f (x) ¯f 0 xf (x)
1/2 ˜f
Applications to standard distributions like Student’s t
[Devroye, 1986, Section 3.7]](https://image.slidesharecdn.com/ratiou-161024122138/85/Ratio-of-uniforms-and-beyond-6-320.jpg)
Ratio of uniforms and beyond
- 1. Ratio of uniforms and beyond Christian P. Robert Universit´e Paris-Dauphine, Paris & University of Warwick, Coventry October 24, 2016
- 2. Ratio of uniforms: what? Consider the set A of (u, v)’s in R+ × X such that 0 u2 f (v/u) Then a uniform distribution on A induces the distribution with density proportional to f on V /U. [Kinderman and Monahan’s (1977)]
- 3. Ratio of uniforms: why? Consider the change of variables from (u, v) to (u, w = v/u) with Jacobian u, then (u, w) has the density u I(0,f (w)1/2)(u) Integrating out u leads to f (w) 1/2 0 u du = f (w) 1/2×2 = f (w) as proportional to the density of V /U
- 4. Ratio of uniforms: how? Simulating a uniform distribution on A means identifying the region within a simple box B Boundaries of A given by (?) Ab = {(u(x) = f (x) 1/2 , v(x) = xf (x) 1/2 ); x ∈ X}
- 5. Ratio of uniforms: how? Simulating a uniform distribution on A means identifying the region within a simple box B Boundaries of A given by (?) Ab = {(u(x) = f (x) 1/2 , v(x) = xf (x) 1/2 ); x ∈ X}
- 6. Ratio of uniforms: why? There exists a compact box B containing A iff 0 f (x) ¯f 0 xf (x) 1/2 ˜f Applications to standard distributions like Student’s t [Devroye, 1986, Section 3.7]
- 7. Ratio of uniforms: where? Principle that can be generalised to a monotone transform of f , h(f ), and the set H = {(u, v); 0 u h(f (v/g(u)))} which still produces a distribution with density proportional to f when g(x) = dG/dx(x) G(x) = h−1 (x)
- 8. Ratio of uniforms: when? choice of transform f most adequate for a given f slice sampler deduced from this construct case of an unbounded density f