1. Nathan Baker (baker@biochem.wustl.edu) BME 540
The Monte Carlo method
Nathan Baker
BME 540

2. What is Monte Carlo? A method to evaluate integrals
Popular in all areas of science, economics, sociology, etc.
Involves:
Random sampling of domain
Evaluation of function
Acceptance/rejection of points

3. Why Monte Carlo? The domain of integration is very complex:
Find the integral of a complicated solvent-solute energy function outside the boundary of a protein.
Find the volume of an object from a tomographic data set.
The integrals are very high-dimensional:
A partition function or free energy for a protein
Analyzing metabolic networks
Predicting trends in the stock market

4. Basic Monte Carlo Algorithm
Suppose we want to approximate
in a high-dimensional space
For i = 1 to n
Pick a point xi at random
Accept or reject the point based on criterion
If accepted, then add f(xi) to total sum
Error estimates are “free” by calculating sums of squares
Error typically decays as

5. Example: the area of a circle
Sample points randomly from square surrounding circle of radius 5
10,000 sample points
Acceptance criterion: inside circle
Actual area:
Calculated area:

6. Example: more complicated shapes

7. Example: multiple dimensions
What is the average of a variable for a N-dimensional probability distribution?
Two approaches:
Quadrature
Discretize each dimension into a set of n points
Possibly use adaptivity to guide discretization
For a reasonably smooth function, error decreases as n-N/2
Monte Carlo
Sample m points from the space
Possibly weight sampling based on reference function
Error decreases as m-1/2

8. Problems: sampling tails of distributions
We want to
Integrate a sharply-peaked function
Use Monte Carlo with uniformly-distributed random numbers
What happens?
Very few points contribute to the integral (~9%)
Poor computational efficiency/convergence
Can we ignore the tails?
NO!
Solution: use a different distribution

9. Improved sampling: change of variables
One way to improve sampling is to change variables:
New distribution is flatter
Uniform variates more useful
Advantages:
Simplicity
Very useful for generating distributions of non-uniform variates (coming up)
Disadvantages
Most useful for invertible functions

10. Change of variables: method
Given an integral
Transform variables
Choose variables to give (nearly) flat distribution
Integrate

11. Change of variables application: exponential
Given an integral
Transform variables
Choose variables to give (nearly) flat distribution
Integrate

12. Change of variables application: exponential
Before transformation:
0.5% of points in domain contribute to integral
Slow convergence

13. Change of variables example: exponential
Before transformation:
0.5% of points in domain contribute to integral
Slow convergence
After transformation:
All of the points in domain contribute
Rapid (exact) convergence
In practice:
Speed-up best when inversion is nearly exact

14. Importance sampling Functions aren’t always invertible
Is there another way to improve sampling of “important” regions of the functions?
Find flat distributions
Bias sampling
Find a function that is almost proportional to the one of interest:
Rewrite your integral as a “weighted” integral:

15. Importance sampling example: a lumpy Gaussian
Our original integrand is
This is close to
Therefore:
Sample random numbers from (we’ll talk about how to do this later):
Evaluate the following integrand over the random number distribution:

16. Importance sampling example: a lumpy Gaussian
Convergence is pretty good (actual value …)

17. Evolution of Monte Carlo methods so far…
Uniform points and original integrand…
…but this had very poor efficiency….
Uniform points and transformed integrand…
…but this only worked for certain integrands….
Non-uniform points and scaled integrand…
…but this is very cumbersome for complicated integrands…
Now, we try Markov chain approaches…

18. Markov chains Properties Examples A sequence of randomly-chosen states
The probability of transitions between states is independent of history
The entire chain represents a stationary probability distribution
Examples
Random number generators
Brownian motion
Hidden Markov Models
(Perfectly) encrypted data

19. Detailed balance
What sorts of Markov chains reach stationary distributions?
This is the same question as posed for the Liouville equation…
Equilibrium processes have stationary distributions
What does it mean to be at equilibrium?
Reversibility
Detailed balance relates stationary probabilities to transitions:

20. Detailed balance: example
Markov chain with a Gaussian stationary probability distribution
Detailed balance satisfied

21. Detailed balance: example
Markov chain with a Gaussian stationary probability distribution, EXCEPT:
Steps to the right are 1% more favorable
Detailed balance violated
When both states are unfavorable, a 1% bias makes a big difference!

22. Markov chain Monte Carlo
Assembling the entire distribution for MC is usually hard:
Complicated energy landscapes
High-dimensional systems
Extraordinarily difficult normalization
Solution:
Build up distribution from Markov chain
Choose local transition probabilities which generate distribution of interest (i.e., ensure detailed balance)
Each random variable is chosen based on the previous variable in the chain
“Walk” along the Markov chain until convergence reached
Result: Normalization not required, calculations are local

23. Application: microcanonical ensemble
Consider the particle in a N-D box.
Monte Carlo algorithm for microcanonical ensemble:
Sample
Determine the number of states with energy ≤E
Relate to the microcanonical partition function by differentiation

24. Application: microcanonical ensemble
9-dimensional particle in a box

25. Markov chain Monte Carlo: flavors
Molecular simulation: Metropolis
Bayesian inferences: Gibbs
Hidden Markov Models: Viterbi
…etc….

26. Application: stochastic transitions
This is a very simplistic version of kinetic Monte Carlo…
Each Monte Carlo step corresponds to a time interval
The probability of moving between states in that time interval is related to the rate constants
Simulate to give mean first passage times, transient populations, etc. A B

27. Metropolis Monte Carlo
Start with the detailed balance condition
Derive an “acceptance ratio” condition
Choose a particular acceptance ratio

28. Application: canonical ensemble
Our un-normalized probabilities look like Boltzmann factors:
Our acceptance ratio is therefore:

29. Algorithm: NVT Metropolis Monte Carlo
Metropolis MC in a harmonic potential

30. Advantages of Metropolis MC simulations
Does not require forces
Rapidly-changing energy functions
No differentiation required
Amenable to complex move sets:
Torsions
Rotamers
Tautomers
Etc.

31. Monte Carlo “machinery”
Boundary conditions
Finite
Periodic
Interactions
Complete
Truncated
How do we choose “moves”?

32. Monte Carlo moves Trial moves Questions: Rigid body translation
Rigid body rotation
Internal conformational changes (soft vs. stiff modes)
Titration/electronic states …
Questions:
How “big” a move should we take?
Move one particle or many?

33. Monte Carlo moves How “big” a move should we take?
Smaller moves: better acceptance rate, slower sampling
Bigger moves: faster sampling, poorer acceptance rate
Amortize mean squared displacement with respect to CPU time
“Rules of thumb”: percent acceptance rate
Move one particle or many?
Possible to achieve more efficient sampling with correct multi-particle moves
One-particle moves must choose particles at random

34. Random variates Don’t develop your own uniform variate method
Look for long period, lack of bias, etc.
These are usually fulfilled in random(), drand48(), etc.
I like “Mersenne twister”
Use uniform variates for starting point of other distributions
Can use all the methods we’ve described:
Standard sampling (rejection method)
Transformations
Metropolis (watch out for correlations)

35. Random variates: transformations
Useful for functions with easy inverses
Sample uniform variable and transform via inverse to give particular variates
Tougher distributions require rejection with respect to “easy” distribution…

36. Problem: poor convergence
Error in Monte Carlo decreases as N-1/2
What if you want to improve error by 1%?
This is due to the “filling behavior” of uniformly-distributed random points
Affects nearly all areas of MC simulations
Convergence can also be difficult to detect:
Observe decrease in variance of data
Employ error estimates
Sub-sample data

37. Quasi-random sequences
Compare to the “pseudo-random” methods we’ve discussed so far.
Can we find a sub-random sequence with low internal correlations that fills space better?
Solution: a maximally-avoiding set of points
Number-theoretic methods:
Base transformations
Primes
Polynomials
Convergence can reach N-1 (Sobol sequence)
Caveats:
How many points (resolution)?
What about sharply-varying functions?
Hammersley quasi-random sequence

38. Application: the bootstrap method
Useful for estimating error fit parameters
Resampling of original data
Examination of resulting fit parameters
Algorithm:
Generate fit parameters for the original data
For (number of resamples desired)
Resample the data with replacement
Generate new fit parameters
Save deviation with respect to original fit parameters
Analyze deviation statistics

39. Summary A method for sampling
Integrals
Configuration space
Error space
Any distribution that can be represented by transitions can be sampled
Sampling can be accelerated using various tricks