Princeton University Department of Geosciences Course on Inverse Problems Albert Tarantola Fourth Lesson: Sampling a Probability Distribution
Definition: a randomly generated point P is a sample point of the volumetric probability f ( P ) if the probability that P happens to be inside any domain A is the probability of A , i.e., if the probability of P ∈ A is � P [ A ] = A dV f ( P ) .
Sampling a volumetric probability f ( P ) : 1 take some value F ≥ f max (if possible, F = f max ); 2 generate a sample point P of the homogeneous distribu- tion; 3 give a chance to point P of being accepted, with proba- bility of acceptance P = f ( P ) / F ; 4 if P is rejected, go to 2, and so until a point is accepted. Theorem: When a point P is accepted, it is a sample point of the volumetric probability f ( P ) .
400 300 200 100 0 -7.5 -5 -2.5 0 2.5 5 7.5
⇒ mathematica notebook
4 2 0 � 2 � 4 � 2 0 2 4 6 8
⇒ mathematica notebook
⇒ mathematica notebook
Metropolis algorithm for sampling a volum. proba. f ( P ) 1 Design a random walk P i , P i + 1 , . . . that, if unthwarted, samples the homogeneous distribution; 2 let P current be the last accepted point, and P test the next point that the homogeneous random walk proposes; 3 if f ( P test ) ≥ f ( P current ) , accept P test as the new current point; 4 if f ( P test ) < f ( P current ) , give a chance to point P test of being accepted, with probability of acceptance P = f ( P test ) / f ( P current ) . Theorem: The sequence of accepted points is, asymptotically, a sequence of sample points of the volumetric probability f ( P ) .
Example: Sampling a 10-dimensional volumetric probability using the Metropolis algorithm. ( ⇒ mathematica notebook).
Volume hypersphere: 2 π n / 2 r n n Γ ( n / 2 ) Volume hypercube: ( 2 r ) n Volume hypersphere / Volume hypercube 1 0.8 0.6 0.4 0.2 0 1 2 3 4 5 6 7 8 9 10 11 Dimension
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