Nested Sampling with Constrained Hamiltonian Monte Carlo Michael Betancourt Massachusetts Institute of Technology MaxEnt2010

L |α, (α)M) pπ(α|M) (α) p (D p (α|D , M) = p (DZ|M)

Z1 π (M1) p (M1|D ) log = log p (M1|D ) Z2 π (M2) p (α|D ) ∝ ∑ p (α|D , Mi) Zi π (Mi) i

Z=

Z

m

d α L (α) π (α)

Nested Sampling

α˜ = {α|L (α) > L}

x (L) =

Z

α˜

m

d α π (α)

x=1

x=0 L=0

L = Lmax

dx (L) =

dx (L) =

Z

m

∂α˜

Z

∂α˜

d α π (α) m−1

dα⊥d

dx (L) = dα⊥

Z

∂α˜

d

α# π (α)

m−1

dx (L) = dα⊥ π (α⊥)

α# π (α)

Z= Z= Z= Z=

Z

Z

Z

Z

m

d α L (α) π (α) m−1

dα⊥d

α# L (α) π (α)

dα⊥ L (α⊥) π (α⊥) dx L (x)

!

1, 0 ≤ xn−1 ≤1 π (x) p (x=max)0,=otherwise n x max x=0

x=1

(x1, L1) = (x˜max, Lmin)

!

"n−1

! n x max 1, 0 ≤ x ≤ x π (α) , L (α) > L 1 1 π˜ (x) ∝ p (xmax|x1) =π˜ (α) ∝ 0, otherwise x1 x1 0, otherwise

!

x1

x=0

x=1

(x2, L2) = (x˜max, Lmin)

(x1, L1) (x2, L2) (x3, L3) (x4, L4) (x5, L5)

!

π (α) , L (α) > L π˜ (α) ∝ 0, otherwise

!

π (α) , L (α) > L π˜ (α) ∝ 0, otherwise

Hamiltonian Monte Carlo

! " 1 2 p (x) ∝ exp [−E (x)] , p (p) ∝ exp − |p| 2

}

! " #$ 1 2 p (x, p) ∝ exp − |p| + E (x) 2

H

dx ∂H = =p dt ∂p dp ∂H =− = −∇E (x) dt ∂x

! " ! ! p ∼ p p |x = N (0, I) ! " ! " ! " ! ! ! ∗ ! x ∼ p x |p ∝ π δ x − x + (1 − π) δ x − x ! " #$ H (x, p) π = min 1, exp H (x∗, p∗)

Constrained Hamiltonian Monte Carlo

p (x)

E (x)

p = p − 2 (p · n) nˆ nˆ !

∇C C (x) nˆ = |∇C C (x)|

Nested Sampling with Constrained Hamiltonian Monte Carlo

http://web.mit.edu/~betan/www/code.html

 $ % exp λ , Accept R $ % εi+1 = εi exp λ , Reject 1−R

1 2 1 T −1 |p| + E (x) → p M p + E (x) 2 2