1. Monte Carlo integration
It is a numerical probabilistic algorithm a b
I/(b-a) f
Prabhas Chongstitvatana

2. Prabhas Chongstitvatana
MCint(f,n,a,b)
sum = 0
For i = 1 to n do
x = uniform(a,b)
sum = sum + f(x)
Return ( b-a) * (sum/n)
I/(b-a)
b-a
Prabhas Chongstitvatana

3. Prabhas Chongstitvatana
Variance of the estimate is inverse proportion to n
The expected error is proportion to
A deterministic algorithm for integration will sample at regular interval.
Prabhas Chongstitvatana

4. Prabhas Chongstitvatana
DETint(f,n,a,b)
sum = 0
delta = (b-a)/n
x = a + delta/2
For I = 1 to n do
sum = sum + f(x)
x = x + delta
Return sum * delta
Prabhas Chongstitvatana

5. Prabhas Chongstitvatana
Advantage of MCint when doing multiple integral in high dimension. As the sample point increases exponentially with the dimension.
Example 100, 100x100, 100x100x100. MCint is faster than a deterministic algorithm for dimension >= 4.
Prabhas Chongstitvatana

6. Probabilistic counting
A n-bit register can count up to 2n events (0..2n-1)
init(c) ; reset
tick(c) ; increment
count(c) ; return count value
Skip some value to count farther
Prabhas Chongstitvatana

7. Prabhas Chongstitvatana
tick(c) flip a coin if head increment
count(c) return value x 2
Expected value after call count() t time is exactly t.
Prabhas Chongstitvatana

8. Prabhas Chongstitvatana
tick(c) flip a coin if head increment
count(c) return value x 2
Expected value after call count() t time is exactly t.
Count exponentially farther
(8-bit count more than 5 x 1076 events)
count(c) return 2c – 1
Prabhas Chongstitvatana

9. Prabhas Chongstitvatana
Estimate number of tick
2c+1 – 1 with probability p
2c – 1 with prob. (1- p)
Expected value
(2c+1 – 1 ) p + (2c – 1 ) ( 1 – p ) = 2c + 2c p – 1
p = 2-c will get the correct expected value of count(c)
Prabhas Chongstitvatana

10. Prabhas Chongstitvatana
init(c)
c = 0
tick(c)
For i = 1 to c do
if coinflip = head then return
c = c + 1
count(c)
Return 2c – 1
Prabhas Chongstitvatana

11. Prabhas Chongstitvatana
Variance of count after m ticks is m(m-1)/2. The standard deviation is 70%.
This means it cannot distinguish between one million and two million but it can distinguish between one million and 100 million.
Prabhas Chongstitvatana

12. Prabhas Chongstitvatana
Count logarithmic farther
count(c) =
when e is small
e = 1/30 count up to > events in 8-bit register with less than 24% relative error with confidence 95%.
What is probability that tick(c) increment ?
Prabhas Chongstitvatana