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Exploring the connection between sampling problems in Bayesian inference and statistical mechanics Andrew Pohorille NASA-Ames Research Center Outline • Enhanced sampling of pdfs flat histograms multicanonical method Wang-Landau transition probability method parallel tempering • Dynamical systems • Stochastic kinetics Enhanced sampling techniques Preliminaries define: variables x, , N a function U(x,,N) a probability: marginalize x energies are Boltzmann-distributed = 1/kT partition function Q(x,,N) define “free energy” or “thermodynamic potential” 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 scheme insertion deletion adjust weights free energy insertion deletion adjust free energy Multicanonical method normalization of bin count shift 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 or better • Iterate until P(N; 1) is flat Typical example Wang-Landau sampling Example: estimate entropies for (discrete) states entropy acceptance criterion update constant Wang and Landau, Phys. Rev. Lett. 86, 2050 (2001), Phys. Rev. E 64, 056101 (2001) 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 J I i j K Wang, Tay, Swendsen, Phys. Rev. Lett., 82 476 (1999) Fitzgerald et al. J. Stat. Phys. 98, 321 (1999) detailed balance macroscopic detailed balance Parallel tempering Dynamical systems Assumption -ergodicity The idea: the system evolves according to equations of motion (possibly Hamiltonian) we need to assign masses to variables Advantages • No need to design sampling techniques • Specialized methods for efficient sampling are available (Laio-Parrinello, Adaptive Biasing Force) Disadvantages • No probabilistic sampling • Possibly complications with assignment of masses Two formulations: • Hamiltonian • Lagrangian Numerical, iterative solution of equations of motion (a trajectory) Assignment of masses Energy equipartition needs to be addressed • Masses too large - slow motions • Masses too small - difficult integration of equations of motion • Large separation of masses - adiabatic separation Thermostats are available Lagrangian - e.g. Nose-Hoover Hamiltonian - Leimkuhler Adaptive Biasing Force force A = b ∂H()/∂ d * a * Darve and Pohorile, J. Chem. Phys. 115:9169-9183 (2001). 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 Assumptions • {k} are constant • The process is Markovian (well-stirred reactor) Example 7 objects 5 transformations Deterministic solution kinetics (differential equations) concentrations 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 104 1876 (2000) Extensions • • • • k = k(t) (GB) Non-Markovian processes (GB) Stiff reactions (Eric van den Eijden) Enzymatic reactions (A.P.)