Jan 2014 Intro to Bayesian Probability, Statistical Inference, Sampling
Exploring the connection between sampling problems in Bayesian inference and statistical mechanics
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Transcript of Exploring the connection between sampling problems in Bayesian inference and statistical mechanics
Exploring the connection between sampling problems in Bayesian inference and
statistical mechanics
Andrew Pohorille
NASA-Ames Research Center
Outline
• Enhanced sampling of pdfs
• Dynamical systems
• Stochastic kinetics
flat histogramsmulticanonical methodWang-Landautransition probability methodparallel tempering
Enhanced sampling techniques
Preliminariesdefine: variables x, , N a function U(x,,N) a probability:
marginalize x
define “free energy” or “thermodynamic potential”
partition function Q(x,,N)
energies areBoltzmann-distributed = 1/kT
The problem:
What to do if
is difficult to estimate because we can’t get sufficient statistics for all of interest.
Flat histogram approach
pdf sampled uniformly for all , N
weight
Example:
original pdf
weighted pdf
marginalization
“canonical”partition function
1. get 2. get Q
General MC sampling schemeinsertion
deletion
insertion
deletion
adjust weights
adjust free energy
free energy
Multicanonical method
bin count shift
normalization of
Berg and Neuhaus, Phys. Rev. Lett. 68, 9 (1992)
The algorithm• Start with any weights (e.g. 1(N) = 0)
• Perform a short simulation and measure P(N; 1) as histogram
• Update weights according to
• Iterate until P(N; 1) is flat
or better
Typical example
Wang-Landau sampling
entropy
acceptance criterion
Wang and Landau, Phys. Rev. Lett. 86, 2050 (2001), Phys. Rev. E 64, 056101 (2001)
update constant
Example: estimate entropies for (discrete) states
The algorithm
• Set entropies of all states to zero; set initial g
• Accept/reject according to the criterion:
• Always update the entropy estimate for the end state
• When the pdf is flat reduce g
Transition probability method
i j
K
IJ
Wang, Tay, Swendsen, Phys. Rev. Lett., 82 476 (1999)Fitzgerald et al. J. Stat. Phys. 98, 321 (1999)
detailed balance
macroscopicdetailed balance
Parallel tempering
Dynamical systems
The idea: the system evolves according to equations of motion (possibly Hamiltonian)
we need to assign masses to variables
Assumption -ergodicity
Advantages• No need to design sampling techniques
• Specialized methods for efficient sampling are available (Laio-Parrinello, Adaptive Biasing Force)
• No probabilistic sampling
• Possibly complications with assignment of masses
Disadvantages
Two formulations:
• Hamiltonian
• Lagrangian
Numerical, iterative solution of equations of motion (a trajectory)
Assignment of masses
• Masses too large - slow motions
• Masses too small - difficult integration of equations of motion
• Large separation of masses - adiabatic separation
Thermostats are availableLagrangian - e.g. Nose-HooverHamiltonian - Leimkuhler
Energy equipartition needs to be addressed
Adaptive Biasing Force
A = a
b ∂H()/∂ *d
*
Darve and Pohorile, J. Chem. Phys. 115:9169-9183 (2001).
force A
Summary
• A variety of techniques are available to sample efficiently rarely visited states.
• Adaptive methods are based on modifying sampling while building the solution.
• One can construct dynamical systems to seek the solution and efficient adaptive techniques are available. But one needs to do it carefully.
Stochastic kinetics
The problem• {Xi} objects, i = 1,…N
• ni copies of each objects
• undergo r transformations
• With rates {k}, = 1,…r
• {k} are constant
• The process is Markovian (well-stirred reactor)
Assumptions
Example
7 objects5 transformations
Deterministic solution
concentrations
kinetics (differential equations)
steady state (algebraic equations)
Works well for large {ni} (fluctuations suppressed)
A statistical alternative
generate trajectories
• which reaction occurs next?• when does it occur?
next reaction is at time
next reaction is at any time
any reaction at time
Direct method - Algorithm
• Initialization
• Calculate the propensities {ai}
• Choose (r.n.)
• Choose (r.n.)
• Update no. of molecules and reset tt+ • Go to step 2
Gillespie, J. Chem. Phys. 81, 2340 (1977)
First reaction method -Algorithm• Initialization
• Calculate the propensities {ai}
• For each generate according to (r.n.)
• Choose reaction for which is the shortest
• Set =
• Update no. of molecules and reset tt+ • Go to step 2
Gillespie, J. Chem. Phys. 81, 2340 (1977)
Next reaction method
Complexity - O(log r)
Gibson and Bruck,J. Phys. Chem. A 1041876 (2000)
Extensions
• k = k(t) (GB)
• Non-Markovian processes (GB)
• Stiff reactions (Eric van den Eijden)
• Enzymatic reactions (A.P.)