2. Markov Chain Monte Carlo
• Class of sampling algorithms
• High sampling efficiency
• Sample from a distribution with unknown normalization constant
• Often the only way to solve problems in time polynomial in the
number of dimensions
e.g. evaluation of a convex body volume
4. The Monte Carlo principle
• Sample a set of N independent and identically-distributed variables
• Approximation of the target p.d.f. with the empirical expression
… then approximation of the integrals!
6. Idea
• I can use the previously sampled value to find the following one
• Exploration of the configuration space by means of Markov Chains:
def .: Markov process
def .: Markov chain
7. Invariant distribution
• Stability conditions:
1. Irreducibility= for every state there exists a finite probability to visit
any other state
2. Aperiodicity = there are no loops.
• Sufficient condition
1. Detailed balance principle
MCMC algorithms are aperiodic, irreducible Markov chains having
the target pdf as the invariant distribution
8. Example
• What is the probability to find the lift at the ground floor in a three
floor building?
▫ 3 states Markov chain
▫ Lift= Random Walker
▫ Transition matrix
▫ Looking for the invariant distribution
… burn-in …
9. Example - 2
• I can apply the matrix T on the right to any of the states, e.g.
homogeneous
Markov chain
~ 50% is the probability to find
• Google’s PageRank: the lift at the ground floor
▫ Websites are the states, T is defined by the number of hyperlinks among
them and the user is the random walker:
The webpages are displayed following the invariant distribution!
10. Metropolis-Hastings
• Given the target distribution
equivalent to T
1. Choose a value for
2. Sample from a proposal distribution
3. Accept the new value with probability
4. Return to 1
Ratio independent Equal in Metropolis algorithm
of the normalization!
11. M.-H. – Pros and Cons
• Very general sampling method:
▫ I can sample from a unnormalized distribution
▫ It does not require to provide upper bound for the function
• Good working depends on the choice of the proposal distribution
▫ well-mixing condition
12. M.-H. - Example
• In Statistical Mechanics it is important to evalue the partition
function,
e.g. Ising model
Sum every possible spin state:
In a 10 x 10 x 10 spin cube,
I would have to sum over
MCMC APPROACH:
1. Evaluate the system’s energy Possible states = UNFEASIBLE
2. Pick up a spin at random and flip it:
1. If energy decreases, this is the new spin configuration
2. If energy increases, this is the new spin configuration with
probability
13. Simulated Annealing
• It allows one to find the global maximum of a generic pdf
▫ No comparison between the value of local minima required
▫ Application to the maximum-likelihood method
• It is a non-homogeneous Markov chain whose invariant distribution
keeps changing as follows:
14. Simulated Annealing: example
• Let us apply the algorithm to a simple, 1-dimensional case
• The optimal cooling scheme is
15. Simulated Annealing: Pros and Cons
• The global maximum is univocally determined
▫ Even if walker starts next to a local (non global!) maximum, it converges to the
true global maximum
• It requires a good tuning of the parameters
16. Gibbs Sampler
• Optimal method to marginalize multidimensional distributions
• Let us assume we have a n-dimensional vector and that we know all
the conditional probability expression for the pdf
• We take the following proposal distribution:
19. Gibbs Sampler – practically
1. §Initialize fix n-1 coordinates and sample
from the resulting pdf
2. for (i=0 ; i < N; i++)
• Sample
• Sample
• Sample
• Sample
20. Gibbs Sampler – example
• Let us pretend we cannot determine the normalization
constant…
… but we can make a comparison with the true marginalized
pdf…
21. Gibbs Sampler – results
• Comparison between Gibbs
Sampling and the true M.-H.
sampling from the marginalized pdf
• Good c2 agreement
22. A complex MCMC application
A radioactive source decays with frequency l1 and a detector records
only every k1 –th event, then at the moment tc the decay rate
changes to l2 and only one event out ofk2 is recorded.
Apparently l1 , k1 , tc , l2 and k2 are undetermined.
We wish to find them.
23. Preparation
• The waiting time for the k-th event in a Poissonian process with
frequency l is distributed according to:
• I can sample a big amount of events from this pdf, changing the
parameters l1 e k1 to l2 e k2 at time tc
• I evaluate the likelihood:
24. Idea
• I assume log-likelihood to be the invariant distribution!
▫ which are the Markov chain states?
struct State {
Parameter
double lambda1, lambda2;
space
double tc;
int k1, k2; Corresponding log-
double plog; likelihood value
State(double la1, double la2, double t, int kk1, int kk2) :
lambda1(la1), lambda2(la2), tc(t), k1(kk1), k2(kk2) {}
State() {};
};
25. Practically
• I have to find an appropriate proposal distribution to move among
the states
▫ Attention: varying li and ki I have toi prevent the acceptance rate to be
too low… but also too high!
• The a ratio is evaluated as the ratio between the final-state and
initial-state likelihood values.
• Try to guess the values for li , ki and tc
• Let the chain evolve for a burn-in time and then record the results.
26. Results • Even if the inital guess is quite far from the real
value, the random walker converges.
guess: l1=5 l2 = 5 k1 = 3 k2 = 2
real: l1=1 l2 = 2 k1 = 1, k2 = 1
27. Results- 2
• Estimate of the uncertainty
l2
l1
28. Results- 3
• All the parameters can be detemined quickly
guess: tc=150 real: tc=300
29. References
• C. Andrieu, N. De Freitas, A. Doucet e M.I. Jordan, Machine Learning 50
(2003), 5-43.
• G. Casella e E.I. George, The American Statistician 46, 3 (1992), 167-174.
• W.H. Press, S. A. Teukolsky, W.T. Vetterling e B.P. Flannery, Numerical
Recipes , Third Edition, Cambridge University Press, 2007.
• M. Loreti, Teoria degli errori e fondamenti di statistica, Decibel, Zanichelli
(1998).
• B. Walsh, Markov Chain Monte Carlo and Gibbs Sampling, Lecture Notes
for EEB 581
