1.
Simulating events of unknown probability by
reverse time martingales
(poprzez martyngaly z czasem odwr´conym)
o
Omiros Papaspiliopoulos
UPF, Barcelona
joint work with
Krzysztof Latuszynski Ioannis Kosmidis Gareth O. Roberts
(Warwick University)

2.
Motivation
The Bernoulli factory - general results and previous approaches
Reverse time martingale approach to sampling
Application to the Bernoulli Factory problem

3.
Generic description of the problem
Let p ∈ (0, 1) be unknown.
Given a black box that samples p−coins
Can we construct a black box that samples f (p) coins for known f ?
For example f (p) = min(1, 2p)

4.
Some history
(see for example Peres, 1992)
von Neumann posed and solved the problem: f (p) = 1/2

5.
Some history
(see for example Peres, 1992)
von Neumann posed and solved the problem: f (p) = 1/2
1. set n = 1;
2. sample Xn , Xn+1
3. if (Xn , Xn+1 ) = (0, 1) output 1 and STOP
4. if (Xn , Xn+1 ) = (1, 0) output 0 and STOP
5. set n := n + 2 and GOTO 2.
Lets check why this work

6.
The Bernoulli Factory problem
for known f and unknown p, how to generate an f (p)−coin?
von Neumann: f (p) = 1/2
Asmussen conjectured f (p) = 2p, but it turned out difficult

7.
Exact simulation of diffusions as Bernoulli factory
This is the description of EA closest in spirit to Beskos and Roberts
(2005)
Simulate XT at time T > 0 from:
dXt = α(Xt ) dt + dWt , X0 = x ∈ R, t ∈ [0, T ] (1)
driven by the Brownian motion {Wt ; 0 ≤ t ≤ T }

8.
Ω ≡ C ([0, T ], R), co-ordinate mappings Bt : Ω → R, t ∈ [0, T ], such
that for any t, Bt (ω) = ω(t) and the cylinder σ-algebra
C = σ({Bt ; 0 ≤ t ≤ T }).
W x = {Wtx ; 0 ≤ t ≤ T } the Brownian motion started at x ∈ R, and by
W x,u = {Wtx,u ; 0 ≤ t ≤ T } the Brownian bridge.

9.
1. The drift function α is differentiable.
2. The function h(u) = exp{A(u) − (u − x)2 /2T }, u ∈ R, for
u
A(u) = 0 α(y )dy , is integrable.
3. The function (α2 + α )/2 is bounded below by > −∞, and above
by r + < ∞.
1 2
φ(u) = [(α + α )/2 − ] ∈ [0, 1] , (2)
r

10.
Q be the probability measure induced by the solution X of (1) on (Ω, C),
W the corresponding probability measure for W x , and Z be the
probability measure defined as the following simple change of measure
from W: dW/dZ(ω) ∝ exp{−A(BT )}. Note that a stochastic process
distributed according to Z has similar dynamics to the Brownian motion,
with the exception of the distribution of the marginal distribution at time
T (with density, say, h) which is biased according to A.

11.
Q be the probability measure induced by the solution X of (1) on (Ω, C),
W the corresponding probability measure for W x , and Z be the
probability measure defined as the following simple change of measure
from W: dW/dZ(ω) ∝ exp{−A(BT )}. Note that a stochastic process
distributed according to Z has similar dynamics to the Brownian motion,
with the exception of the distribution of the marginal distribution at time
T (with density, say, h) which is biased according to A. Then,
T
dQ
(ω) ∝ exp −rT T −1 φ(Bt )dt ≤1 Z − a.s. (3)
dZ 0

12.
EA using Bernoulli factory
1. simulate u ∼ h
T
2. generate a Cs coin where s := e −rTJ , and J := 0
T −1 φ(Wtx,u )dt;
3. If Cs = 1 output u and STOP;
4. If Cs = 0 GOTO 1.
Exploiting the Markov property, we can assume from now on that rT < 1.

13.
The challenging part of the algorithm is Step 2, since exact computation
of J is impossible due to the integration over a Brownian bridge path.
On the other hand, it is easy to generate J-coins:
x,u
CJ = I(ψ < φ(Wχ )), ψ ∼ U(0, 1) , χ ∼ U(0, T )
independent of the Brownian bridge W x,u and of each other.
Therefore, we deal with another instance of the problem studied in this
article: given p-coins how to generate f (p)-coins, where here f is the
exponential function.
(Note the interplay between unbiased estimation and exact simulation
here)

14.
Keane and O’Brien - existence result
Keane and O’Brien (1994):
Let p ∈ P ⊆ (0, 1) → [0, 1]
then it is possible to simulate an f (p)−coin ⇐⇒
f is constant
f is continuous and for some n ∈ N and all p ∈ P satisfies
n
min f (p), 1 − f (p) ≥ min p, 1 − p
Note that the result rules out min{1, 2p}, but not min{1 − 2 , 2p}

15.
Keane and O’Brien - existence result
Keane and O’Brien (1994):
Let p ∈ P ⊆ (0, 1) → [0, 1]
then it is possible to simulate an f (p)−coin ⇐⇒
f is constant
f is continuous and for some n ∈ N and all p ∈ P satisfies
n
min f (p), 1 − f (p) ≥ min p, 1 − p
however their proof is not constructive
Note that the result rules out min{1, 2p}, but not min{1 − 2 , 2p}

16.
Nacu-Peres (2005) Theorem - Bernstein polynomial
approach
There exists an algorithm which simulates f ⇐⇒ there exist
polynomials
n n
n n
gn (x, y ) = a(n, k)x k y n−k , hn (x, y ) = b(n, k)x k y n−k
k k
k=0 k=0
with following properties

17.
Nacu-Peres (2005) Theorem - Bernstein polynomial
approach
There exists an algorithm which simulates f ⇐⇒ there exist
polynomials
n n
n n
gn (x, y ) = a(n, k)x k y n−k , hn (x, y ) = b(n, k)x k y n−k
k k
k=0 k=0
with following properties
0 ≤ a(n, k) ≤ b(n, k) ≤ 1

18.
Nacu-Peres (2005) Theorem - Bernstein polynomial
approach
There exists an algorithm which simulates f ⇐⇒ there exist
polynomials
n n
n n
gn (x, y ) = a(n, k)x k y n−k , hn (x, y ) = b(n, k)x k y n−k
k k
k=0 k=0
with following properties
0 ≤ a(n, k) ≤ b(n, k) ≤ 1
n n
k a(n, k) and k b(n, k) are integers

19.
Nacu-Peres (2005) Theorem - Bernstein polynomial
approach
There exists an algorithm which simulates f ⇐⇒ there exist
polynomials
n n
n n
gn (x, y ) = a(n, k)x k y n−k , hn (x, y ) = b(n, k)x k y n−k
k k
k=0 k=0
with following properties
0 ≤ a(n, k) ≤ b(n, k) ≤ 1
n n
k a(n, k) and k b(n, k) are integers
limn→∞ gn (p, 1 − p) = f (p) = limn→∞ hn (p, 1 − p)

20.
Nacu-Peres (2005) Theorem - Bernstein polynomial
approach
There exists an algorithm which simulates f ⇐⇒ there exist
polynomials
n n
n n
gn (x, y ) = a(n, k)x k y n−k , hn (x, y ) = b(n, k)x k y n−k
k k
k=0 k=0
with following properties
0 ≤ a(n, k) ≤ b(n, k) ≤ 1
n n
k a(n, k) and k b(n, k) are integers
limn→∞ gn (p, 1 − p) = f (p) = limn→∞ hn (p, 1 − p)
for all m < n
(x+y )n−m gm (x, y ) gn (x, y ) and (x+y )n−m hm (x, y ) hn (x, y )

21.
Nacu-Peres (2005) Theorem - Bernstein polynomial
approach
There exists an algorithm which simulates f ⇐⇒ there exist
polynomials
n n
n n
gn (x, y ) = a(n, k)x k y n−k , hn (x, y ) = b(n, k)x k y n−k
k k
k=0 k=0
with following properties
0 ≤ a(n, k) ≤ b(n, k) ≤ 1
n n
k a(n, k) and k b(n, k) are integers
limn→∞ gn (p, 1 − p) = f (p) = limn→∞ hn (p, 1 − p)
for all m < n
(x+y )n−m gm (x, y ) gn (x, y ) and (x+y )n−m hm (x, y ) hn (x, y )
Nacu & Peres provide coefficients for f (p) = min{2p, 1 − 2ε}
explicitly.

22.
Nacu-Peres (2005) Theorem - Bernstein polynomial
approach
There exists an algorithm which simulates f ⇐⇒ there exist
polynomials
n n
n n
gn (x, y ) = a(n, k)x k y n−k , hn (x, y ) = b(n, k)x k y n−k
k k
k=0 k=0
with following properties
0 ≤ a(n, k) ≤ b(n, k) ≤ 1
n n
k a(n, k) and k b(n, k) are integers
limn→∞ gn (p, 1 − p) = f (p) = limn→∞ hn (p, 1 − p)
for all m < n
(x+y )n−m gm (x, y ) gn (x, y ) and (x+y )n−m hm (x, y ) hn (x, y )
Nacu & Peres provide coefficients for f (p) = min{2p, 1 − 2ε}
explicitly.
Given an algorithm for f (p) = min{2p, 1 − 2ε} Nacu & Peres
develop a calculus that collapses every real analytic g to nesting
the algorithm for f and simulating g .

23.
too nice to be true?
at time n the N-P algorithm computes sets An and Bn - subsets
of all 01 strings of length n

24.
too nice to be true?
at time n the N-P algorithm computes sets An and Bn - subsets
of all 01 strings of length n
n
the cardinalities of An and Bn are precisely k a(n, k) and
n
k b(n, k)

25.
too nice to be true?
at time n the N-P algorithm computes sets An and Bn - subsets
of all 01 strings of length n
n
the cardinalities of An and Bn are precisely k a(n, k) and
n
k b(n, k)
the upper polynomial approximation is converging slowly to f

26.
too nice to be true?
at time n the N-P algorithm computes sets An and Bn - subsets
of all 01 strings of length n
n
the cardinalities of An and Bn are precisely k a(n, k) and
n
k b(n, k)
the upper polynomial approximation is converging slowly to f
length of 01 strings is 215 = 32768 and above, e.g.
225 = 16777216

27.
too nice to be true?
at time n the N-P algorithm computes sets An and Bn - subsets
of all 01 strings of length n
n
the cardinalities of An and Bn are precisely k a(n, k) and
n
k b(n, k)
the upper polynomial approximation is converging slowly to f
length of 01 strings is 215 = 32768 and above, e.g.
225 = 16777216
25
one has to deal efficiently with the set of 22 strings, of length
225 each.

28.
too nice to be true?
at time n the N-P algorithm computes sets An and Bn - subsets
of all 01 strings of length n
n
the cardinalities of An and Bn are precisely k a(n, k) and
n
k b(n, k)
the upper polynomial approximation is converging slowly to f
length of 01 strings is 215 = 32768 and above, e.g.
225 = 16777216
25
one has to deal efficiently with the set of 22 strings, of length
225 each.
we shall develop a reverse time martingale approach to the
problem

29.
too nice to be true?
at time n the N-P algorithm computes sets An and Bn - subsets
of all 01 strings of length n
n
the cardinalities of An and Bn are precisely k a(n, k) and
n
k b(n, k)
the upper polynomial approximation is converging slowly to f
length of 01 strings is 215 = 32768 and above, e.g.
225 = 16777216
25
one has to deal efficiently with the set of 22 strings, of length
225 each.
we shall develop a reverse time martingale approach to the
problem
we will construct reverse time super- and submartingales that
perform a random walk on the Nacu-Peres polynomial coefficients
a(n, k), b(n, k) and result in a black box that has algorithmic
cost linear in the number of original p−coins

30.
Before giving the most general algorithm, let us think gradually how to
simulate events of unknown probability constructively

31.
Algorithm 1 - randomization
Lemma: Sampling events of probability s ∈ [0, 1] is equivalent to
constructing an unbiased estimator of s taking values in [0, 1]
with probability 1.

32.
Algorithm 1 - randomization
Lemma: Sampling events of probability s ∈ [0, 1] is equivalent to
constructing an unbiased estimator of s taking values in [0, 1]
with probability 1.
ˆ ˆ ˆ
Proof: Let S, s.t. ES = s and P(S ∈ [0, 1]) = 1 be the
ˆ
estimator. Then draw G0 ∼ U(0, 1), obtain S and define a coin
ˆ
Cs := I{G0 ≤ S}.
ˆ s ˆ s
P(Cs = 1) = E I(G0 ≤ S) = E E I(G0 ≤ ˆ) | S = ˆ ˆ
= ES = s.
The converse is straightforward since an s−coin is an unbiased
estimator of s with values in [0, 1].

33.
Algorithm 1 - randomization
Lemma: Sampling events of probability s ∈ [0, 1] is equivalent to
constructing an unbiased estimator of s taking values in [0, 1]
with probability 1.
ˆ ˆ ˆ
Proof: Let S, s.t. ES = s and P(S ∈ [0, 1]) = 1 be the
ˆ
estimator. Then draw G0 ∼ U(0, 1), obtain S and define a coin
ˆ
Cs := I{G0 ≤ S}.
ˆ s ˆ s
P(Cs = 1) = E I(G0 ≤ S) = E E I(G0 ≤ ˆ) | S = ˆ ˆ
= ES = s.
The converse is straightforward since an s−coin is an unbiased
estimator of s with values in [0, 1].
Algorithm 1
1. simulate G0 ∼ U(0, 1);
2. ˆ
obtain S;
3. ˆ
if G0 ≤ S set Cs := 1, otherwise set Cs := 0;
4. output Cs .

34.
Algorithm 2 - lower and upper monotone deterministic
bounds
let l1 , l2 , ... and u1 , u2 , ... be sequences of lower and upper
monmotone bounds for s converging to s, i.e.
li s and ui s.

35.
Algorithm 2 - lower and upper monotone deterministic
bounds
let l1 , l2 , ... and u1 , u2 , ... be sequences of lower and upper
monmotone bounds for s converging to s, i.e.
li s and ui s.
Algorithm 2
1. simulate G0 ∼ U(0, 1); set n = 1;
2. compute ln and un ;
3. if G0 ≤ ln set Cs := 1;
4. if G0 > un set Cs := 0;
5. if ln < G0 ≤ un set n := n + 1 and GOTO 2;
6. output Cs .

36.
Algorithm 2 - lower and upper monotone deterministic
bounds
let l1 , l2 , ... and u1 , u2 , ... be sequences of lower and upper
monmotone bounds for s converging to s, i.e.
li s and ui s.
Algorithm 2
1. simulate G0 ∼ U(0, 1); set n = 1;
2. compute ln and un ;
3. if G0 ≤ ln set Cs := 1;
4. if G0 > un set Cs := 0;
5. if ln < G0 ≤ un set n := n + 1 and GOTO 2;
6. output Cs .
Remark: P(N > n) = un − ln .

37.
This is a practically useful technique, suggested for example in Devroye
(1986), and implemented for simulation from random measures in
P+Roberts (2008).

38.
Algorithm 3 - monotone stochastic bounds
Ln ≤ Un (4)
Ln ∈ [0, 1] and Un ∈ [0, 1] (5)
Ln−1 ≤ Ln and Un−1 ≥ Un (6)
E Ln = ln s and E Un = un s. (7)
F0 = {∅, Ω}, Fn = σ{Ln , Un }, Fk,n = σ{Fk , Fk+1 , ...Fn } for k ≤ n.

39.
Algorithm 3 - monotone stochastic bounds
Ln ≤ Un (4)
Ln ∈ [0, 1] and Un ∈ [0, 1] (5)
Ln−1 ≤ Ln and Un−1 ≥ Un (6)
E Ln = ln s and E Un = un s. (7)
F0 = {∅, Ω}, Fn = σ{Ln , Un }, Fk,n = σ{Fk , Fk+1 , ...Fn } for k ≤ n.
Algorithm 3
1. simulate G0 ∼ U(0, 1); set n = 1;
2. obtain Ln and Un ; conditionally on F1,n−1
3. if G0 ≤ Ln set Cs := 1;
4. if G0 > Un set Cs := 0;
5. if Ln < G0 ≤ Un set n := n + 1 and GOTO 2;
6. output Cs .

40.
Algorithm 3 - monotone stochastic bounds
Ln ≤ Un (4)
Ln ∈ [0, 1] and Un ∈ [0, 1] (5)
Ln−1 ≤ Ln and Un−1 ≥ Un (6)
E Ln = ln s and E Un = un s. (7)
F0 = {∅, Ω}, Fn = σ{Ln , Un }, Fk,n = σ{Fk , Fk+1 , ...Fn } for k ≤ n.
Algorithm 3
1. simulate G0 ∼ U(0, 1); set n = 1;
2. obtain Ln and Un ; conditionally on F1,n−1
3. if G0 ≤ Ln set Cs := 1;
4. if G0 > Un set Cs := 0;
5. if Ln < G0 ≤ Un set n := n + 1 and GOTO 2;
6. output Cs .

41.
Algorithm 3 - monotone stochastic bounds
Lemma
Assume (4), (5), (6) and (7). Then Algorithm 3 outputs a valid s−coin.
Moreover the probability that it needs N > n iterations equals un − ln .
Proof.
Probability that Algorithm 3 needs more then n iterations equals
E(Un − Ln ) = un − ln → 0 as n → ∞. And since 0 ≤ Un − Ln is a
decreasing sequence a.s., we also have Un − Ln → 0 a.s. So there exists a
ˆ
random variable S, such that for almost every realization of sequences
{Ln (ω)}n≥1 and {Un (ω)}n≥1 we have Ln (ω) ˆ
S(ω) and Un (ω) ˆ
S(ω).
ˆ
By (5) we have S ∈ [0, 1] a.s. Thus for a fixed ω the algorithm outputs
ˆ ˆ
an S(ω)−coin a.s (by Algorithm 2). Clearly E Ln ≤ E S ≤ E Un and
ˆ
hence E S = s.

42.
Algorithm 4 - reverse time martingales
Ln ≤ Un
Ln ∈ [0, 1] and Un ∈ [0, 1]
Ln−1 ≤ Ln and Un−1 ≥ Un
E Ln = ln s and E Un = un s.
F0 = {∅, Ω}, Fn = σ{Ln , Un }, Fk,n = σ{Fk , Fk+1 , ...Fn } for k ≤ n.
The final step is to weaken 3rd condition and let Ln be a reverse time
supermartingale and Un a reverse time submartingale with respect to
Fn,∞ . Precisely, assume that for every n = 1, 2, ... we have
E (Ln−1 | Fn,∞ ) = E (Ln−1 | Fn ) ≤ Ln a.s. and (8)
E (Un−1 | Fn,∞ ) = E (Un−1 | Fn ) ≥ Un a.s. (9)

43.
Algorithm 4 - reverse time martingales
Algorithm 4
˜
1. simulate G0 ∼ U(0, 1); set n = 1; set L0 ≡ L0 ≡ 0 and
˜
U0 ≡ U0 ≡ 1
2. obtain Ln and Un given F0,n−1 ,
3. compute L∗ = E (Ln−1 | Fn ) and Un = E (Un−1 | Fn ).
n
∗
4. compute
˜ ˜ Ln − L∗ ˜
n ˜
Ln = Ln−1 + ∗ Un−1 − Ln−1 (10)
Un − L∗n
˜ ˜ U ∗ − Un ˜ ˜
Un = Un−1 − n Un−1 − Ln−1 (11)
∗
Un − L∗ n
5. ˜
if G0 ≤ Ln set Cs := 1;
6. ˜
if G0 > Un set Cs := 0;
7. ˜ ˜
if Ln < G0 ≤ Un set n := n + 1 and GOTO 2;
8. output Cs .

44.
Algorithm 4 - reverse time martingales
Theorem
Assume (4), (5), (7), (8) and (9). Then Algorithm 4 outputs a valid
s−coin. Moreover the probability that it needs N > n iterations equals
un − ln .
˜ ˜
We show that L and U satisfy (4), (5), (7) and (6) and hence
Algorithm 4 is valid since Algorithm 3 was valid.
In fact, we have constructed a mean preserving transformation, in the
˜ ˜
sense E[Ln ] = E[Ln ], and E[Un ] = E[Un ]. Therefore, the proof is based
on establishing this property and appealing to Algorithm 3.

45.
By construction L0 = 0, U0 = 1 a.s thus L∗ = 0, U0 = 1
0
∗

46.
By construction L0 = 0, U0 = 1 a.s thus L∗ = 0, U0 = 1
0
∗
˜ ˜
Therefore, L1 = L1 , and U1 = U1 (intuitive)

47.
By construction L0 = 0, U0 = 1 a.s thus L∗ = 0, U0 = 1
0
∗
˜ ˜
Therefore, L1 = L1 , and U1 = U1 (intuitive)
Thus,
˜ L2 − L∗
2
L2 = L1 + ∗ (U1 − L1 )
U2 − L∗
2

48.
By construction L0 = 0, U0 = 1 a.s thus L∗ = 0, U0 = 1
0
∗
˜ ˜
Therefore, L1 = L1 , and U1 = U1 (intuitive)
Thus,
˜ L2 − L∗
2
L2 = L1 + ∗ (U1 − L1 )
U2 − L∗
2
Take conditional expectation given F2
Therefore result holds for n = 1, 2, then use induction (following the
same approach)

49.
Unbiased estimators
Theorem
Suppose that for an unknown value of interest s ∈ R, there exist a
constant M < ∞ and random sequences Ln and Un s.t.
P(Ln ≤ Un ) = 1 for every n = 1, 2, ...
P(Ln ∈ [−M, M]) = 1 and P(Un ∈ [−M, M]) = 1 for every n = 1, 2, ...
E Ln = ln s and E U n = un s
E (Ln−1 | Fn,∞ ) = E (Ln−1 | Fn ) ≤ Ln a.s. and
E (Un−1 | Fn,∞ ) = E (Un−1 | Fn ) ≥ Un a.s.
Then one can construct an unbiased estimator of s.
Proof.
After rescaling, one can use Algorithm 4 to sample events of probability
(M + s)/2M, which gives an unbiased estimator of (M + s)/2M and
consequently of s.

50.
A version of the Nacu-Peres Theorem
An algorithm that simulates a function f on P ⊆ (0, 1) exists if and only
if for all n ≥ 1 there exist polynomials gn (p) and hn (p) of the form
n n
n n
gn (p) = a(n, k)p k (1−p)n−k and hn (p) = b(n, k)p k (1−p)n−k ,
k k
k=0 k=0
s.t.
(i) 0 ≤ a(n, k) ≤ b(n, k) ≤ 1,
(ii) limn→∞ gn (p) = f (p) = limn→∞ hn (p),
(iii) For all m < n, their coefficients satisfy
k n−m m k n−m m
k−i i k−i i
a(n, k) ≥ n a(m, i), b(n, k) ≤ n b(m, i). (12)
i=0 k i=0 k

51.
Algorithm 4 - reverse time martingales
Proof: polynomials ⇒ algorithm.
Let X1 , X2 , . . . iid tosses of a p−coin.

52.
Algorithm 4 - reverse time martingales
Proof: polynomials ⇒ algorithm.
Let X1 , X2 , . . . iid tosses of a p−coin.
Define {Ln , Un }n≥1 as follows:

53.
Algorithm 4 - reverse time martingales
Proof: polynomials ⇒ algorithm.
Let X1 , X2 , . . . iid tosses of a p−coin.
Define {Ln , Un }n≥1 as follows:
n
if i=1 Xi = k, let Ln = a(n, k) and Un = b(n, k).

54.
Algorithm 4 - reverse time martingales
Proof: polynomials ⇒ algorithm.
Let X1 , X2 , . . . iid tosses of a p−coin.
Define {Ln , Un }n≥1 as follows:
n
if i=1 Xi = k, let Ln = a(n, k) and Un = b(n, k).
In the rest of the proof we check that (4), (5), (7), (8) and (9) hold
for {Ln , Un }n≥1 with s = f (p). Thus executing Algorithm 4 with
{Ln , Un }n≥1 yields a valid f (p)−coin.
Clearly (4) and (5) hold due to (i). For (7) note that
E Ln = gn (p) f (p) and E Un = hn (p) f (p).

55.
Algorithm 4 - reverse time martingales
Proof - continued.
To obtain (8) and (9) define the sequence of random variables Hn
to be

56.
Algorithm 4 - reverse time martingales
Proof - continued.
To obtain (8) and (9) define the sequence of random variables Hn
to be
n
the number of heads in {X1 , . . . , Xn }, i.e. Hn = i=1 Xi

57.
Algorithm 4 - reverse time martingales
Proof - continued.
To obtain (8) and (9) define the sequence of random variables Hn
to be
n
the number of heads in {X1 , . . . , Xn }, i.e. Hn = i=1 Xi
and let Gn = σ(Hn ). Thus
Ln = a(n, Hn ) and Un = b(n, Hn ), hence Fn ⊆ Gn and it is
enough to check that E(Lm |Gn ) ≤ Ln and E(Um |Gn ) ≥ Un for m < n.

58.
Algorithm 4 - reverse time martingales
Proof - continued.
To obtain (8) and (9) define the sequence of random variables Hn
to be
n
the number of heads in {X1 , . . . , Xn }, i.e. Hn = i=1 Xi
and let Gn = σ(Hn ). Thus
Ln = a(n, Hn ) and Un = b(n, Hn ), hence Fn ⊆ Gn and it is
enough to check that E(Lm |Gn ) ≤ Ln and E(Um |Gn ) ≥ Un for m < n.
The distribution of Hm given Hn is hypergeometric and
Hn n−m m
Hn −i i
E(Lm |Gn ) = E(a(m, Hm )|Hn ) = n a(m, i) ≤ a(n, Hn ) = Ln .
i=0 Hn
Clearly the distribution of Hm given Hn is the same as the
distribution of Hm given {Hn , Hn+1 , . . . }. The argument for Un is
identical.

59.
Practical issues for Bernoulli factory
Given a function f , finding polynomial envelopes satisfying properties
required is not easy. Section 3 of N-P provides explicit formulas for
polynomial envelopes of f (p) = min{2p, 1 − 2ε} that satisfy conditions,
precisely a(n, k) and b(n, k) satisfy (ii) and (iii) and one can easily
compute n0 = n0 (ε) s.t. for n ≥ n0 condition (i) also holds, which is
enough for the algorithm (however n0 is substantial, e.g. n0 (ε) = 32768
for ε = 0.1 and it increases as ε decreases).
The probability that Algorithm 4 needs N > n inputs equals
hn (p) − gn (p). The polynomials provided in N-P satisfy
hn (p) − gn (p) ≤ C ρn for p ∈ [0, 1/2 − 4ε] guaranteeing fast convergence,
and hn (p) − gn (p) ≤ Dn−1/2 elsewhere. Using similar techniques one can
establish polynomial envelopes s.t. hn (p) − gn (p) ≤ C ρn for
p ∈ [0, 1] (1/2 − (2 + c)ε, 1/2 − (2 − c)ε).

60.
Moreover, we note that despite the fact that the techniques developed in
N-P for simulating a real analytic g exhibit exponentially decaying tails,
they are often not practical. Nesting k times the algorithm for
f (p) = min{2p, 1 − 2ε} is very inefficient. One needs at least n0 (ε)k of
original p−coins for a single output.
Nevertheless, both algorithms, i.e. the original N-P and our martingale
modification, use the same number of original p−coins for a single
f (p)−coin output with f (p) = min{2p, 1 − 2ε} and consequently also for
simulating any real analytic function using methodology of N-P. A
significant improvement in terms of p−coins can be achieved only if the
monotone super/sub-martingales can be constructed directly and used
along with Algorithm 3. This is discussed in the next subsection.

61.
Bernoulli Factory for alternating series expansions
Proposition
Let f : [0, 1] → [0, 1] have an alternating series expansion
∞
f (p) = (−1)k ak p k with 1 ≥ a0 ≥ a1 ≥ . . .
k=0
Then an f (p)−coin can be simulated by Algorithm 3 and the probability
that it needs N > n iterations equals an p n .

62.
Proof.
Let X1 , X2 , . . . be a sequence of p−coins and define
U0 := a0 L0 := 0,
n
Un−1 − an k=1 Xk if n is odd,
Ln :=
Ln−1 if n is even,
Un−1 if n is odd,
Un := n
Ln−1 + an k=1 Xk if n is even.
Clearly (4), (5), (6) and (7) are satisfied with s = f (p). Moreover,
un − ln = E Un − E Ln = an p n ≤ an .
Thus if an → 0, the algorithm converges for p ∈ [0, 1], otherwise for
p ∈ [0, 1).

63.
Proof.
Let X1 , X2 , . . . be a sequence of p−coins and define
U0 := a0 L0 := 0,
n
Un−1 − an k=1 Xk if n is odd,
Ln :=
Ln−1 if n is even,
Un−1 if n is odd,
Un := n
Ln−1 + an k=1 Xk if n is even.
Clearly (4), (5), (6) and (7) are satisfied with s = f (p). Moreover,
un − ln = E Un − E Ln = an p n ≤ an .
Thus if an → 0, the algorithm converges for p ∈ [0, 1], otherwise for
p ∈ [0, 1).
The exponential function is precisely in this family

64.
Some References
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Probab. 15(4): 2422–2444, 2005.
A. Beskos, O. Papaspiliopoulos, G.O. Roberts. Retrospective exact
simulation of diffusion sample paths with applications. Bernoulli 12:
1077–1098, 2006.
A. Beskos, O. Papaspiliopoulos, G.O. Roberts, and P. Fearnhead.
Exact and computationally efficient likelihood-based estimation for
discreetly observed diffusion processes (with discussion). Journal of
the Royal Statistical Society B, 68(3):333–382, 2006.
L. Devroye. Non-uniform random variable generation.
Springer-Verlag, New York, 1986.
M.S. Keane and G.L. O’Brien. A Bernoulli factory. ACM
Transactions on Modelling and Computer Simulation (TOMACS),
4(2):213–219, 1994.

65.
P.E. Kloeden and E.Platen. Numerical solution of stochastic
differential equations, Springer-Verlag, 1995.
K. Latuszynski, I. Kosmidis, O. Papaspiliopoulos, G.O. Roberts.
Simulating events of unknown probabilities via reverse time
martingales. Random Structures and Algorithms, to appear.
S. Nacu and Y. Peres. Fast simulation of new coins from old.
Annals of Applied Probability, 15(1):93–115, 2005.
O. Papaspiliopoulos, G.O. Roberts. Retrospective Markov chain
Monte Carlo for Dirichlet process hierarchical models. Biometrika,
95:169–186, 2008.
Y. Peres. Iterating von Neumann’s procedure for extracting random
bits. Annals of Statistics, 20(1): 590–597, 1992.