Choice of an optimal scaling is a common concern when applying Metropolis-Hastings algorithm. We address this problem by suggesting a proposal distribution with state-dependent variance. The new algorithm is generalization of Metropolis-adjusted Langevin algorithm (MALA) and it employs local curvature of target distribution π. Therefore we call it Curvature MALA (CMALA). The motivation for our choice of proposal distribution can be described as follows. Informally speaking if at current state x target distribution is steep we suggest distribution which favours small moves and big leaps are preferred when the chain explores flat parts of π. For one dimensional distributions the proposals of CMALA follow a normal distribution N(μx,σ2x) with σ2x=hk2x and

μx=x+h[12k2x∂∂xlogπ(x)+γkx∂∂xkx],wherekx=∣∣∂2∂x2logπ(x)∣∣−12,γ=1

and h is a time discretization step of Langevin diffusion. Our main interest lies in showing when CMALA and netCMALA (with γ=0) are geometrically ergodic (GE). We obtain that under the regularity conditions

(R0)π(x)∝exp{−|x|β+} for x>M, where M<∞,

and π(x)∝exp{−|x|β−} for x0∀x∈R,(R2∗)π∈C2,(R3)∂2∂x2logπ(x)≠0∀x∈R

and if

(C1.1)min{β+,β−}>1,(C2)h>0 is sufficiently small

hold then netCMALA is well-defined and GE. If all above conditions are satisfied and in addition

(R2)π∈C3

then also CMALA is well-defined and GE. Suppose (R0), (R1), (R2*) and (R3) hold. If also

(C1.2)β+≠β−,min{β+,β−}∈(0,1)

then netCMALA is well-defined but not GE. If in addition (R2) holds then also CMALA is well-defined but not GE. Furthermore we show similar results for CMALA based algorithms adopting randomized h.

Joint work with Gareth O. Roberts, Krzysztof Latuszynski