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Discussion of Matti Vihola's talk
 

Discussion of Matti Vihola's talk

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Presented at Adap'skiii, Park City, Utah, Jan. 3-4, 2010

Presented at Adap'skiii, Park City, Utah, Jan. 3-4, 2010

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    Discussion of Matti Vihola's talk Discussion of Matti Vihola's talk Presentation Transcript

    • On stability and convergence of adaptive MC[MC] Discussion by Christian P. Robert Universit´ Paris-Dauphine, IuF, and CREST e http://xianblog.wordpress.comAdap’skiii, Park City, Utah, Jan. 03, 2011 C.P. Robert On stability and convergence of adaptive MC[MC]
    • The adaptive algorithm Focussing on the scale matrix of an adaptive random-walk Metropolis Hastings scheme is theoretically fascinating, especially when considering removal of containment for LLN [intuitive but hard to prove] potential link with renewal theory [in the fixed component version] no lower bound on Σm verifiable assumptions [Roberts & Rosenthal, 2007] C.P. Robert On stability and convergence of adaptive MC[MC]
    • The adaptive algorithm (2) Limited applicability of the adaptive scheme when using a random-walk Metropolis Hastings scheme because of strong tail [or support] conditions more generaly, dependence on parameterisation curse of dimensionality [ellip. symm. less & less appropriate] unavailability of the performance target [e.g., acceptance of 0.234] C.P. Robert On stability and convergence of adaptive MC[MC]
    • The adaptive algorithm (2) Limited applicability of the adaptive scheme when using a random-walk Metropolis Hastings scheme because of strong tail [or support] conditions more generaly, dependence on parameterisation curse of dimensionality [ellip. symm. less & less appropriate] unavailability of the performance target [e.g., acceptance of 0.234] Congratulations and thanks for adding a software to the theoretical development! C.P. Robert On stability and convergence of adaptive MC[MC]
    • Initialising importance sampling Alternative: Population Monte Carlo (PMC) offers a solution to the difficulty of picking the importance function q through adaptivity: Given a target π, PMC produces a sequence q t of importance functions (t = 1, . . . , T ) aimed at improving the approximation of π [Capp´, Douc, Guillin, Marin & X., 2007] e C.P. Robert On stability and convergence of adaptive MC[MC]
    • Adaptive importance sampling Use of mixture densities D q (x) = q(x; α , θ ) = t t t αd ϕ(x; θd ) t t d=1 [West, 1993] where αt = (α1 , . . . , αD ) is a vector of adaptable weights for the D t t mixture components θt = (θ1 , . . . , θD ) is a vector of parameters which specify the t t components ϕ is a parameterised density (usually taken to be multivariate Gaussian or Student-t) C.P. Robert On stability and convergence of adaptive MC[MC]
    • PMC updates Maximization of Lt (α, θ) leads to closed form solutions in exponential families (and for the t distributions) For instance for Np (µd , Σd ): αd = t+1 ρd (x; αt , µt , Σt )π(x)dx, xρd (x; αt , µt , Σt )π(x)dx µt+1 = d , αd t+1 (x − µt+1 )(x − µt+1 )T ρd (x; αt , µt , Σt )π(x)dx Σt+1 = d d d . αd t+1 C.P. Robert On stability and convergence of adaptive MC[MC]
    • Banana benchmark Twisted Np (0, Σ) target with Σ = diag(σ1 , 1, . . . , 1), changing the 2 second co-ordinate x2 to x2 + b(x1 − σ1 ) 2 2 20 10 0 −10 x2 −20 −30 −40 −40 −20 0 20 40 x1 p = 10, σ1 = 100, b = 0.03 2 [Haario et al. 1999] C.P. Robert On stability and convergence of adaptive MC[MC]
    • Simulation C.P. Robert On stability and convergence of adaptive MC[MC]
    • Comparison to MCMC Adaptive MCMC: Proposal is a multivariate Gaussian with Σ updated/based on previous values in the chain. Scale and update times chosen for optimal results. [Kilbinger, Wraith et al., 2010] PMC MCMC fa fa fb fb Evolution of π(fa ) (top panels) and π(fb ) (bottom panels) from 10k points to 100k points for both PMC (left panels) and MCMC (right panels). C.P. Robert On stability and convergence of adaptive MC[MC]