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Random-walk sampler state

The state of a random-walk Metropolis sampler is encapsulated in the data structure RWMState, which can be constructed as follows:

AdaptiveMCMC.RWMState — Type
s = RWMState(x0, [rng, [q!]])

Constructor for RWMState (random walk Metropolis state).

Arguments

  • x0: The initial state vector
  • rng: Random number generator; default Random.GLOBAL_RNG.
  • q!: Symmetric proposal distribution; default randn!. Called by q!(rng, u) which puts a draw to vector u

If s is RWMState, then s.x is the current state. New proposal may be drawn to s.y by calling draw!, and the proposal is accepted by calling accept!(s).

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By default, the RWMState will provide a Gaussian random walk sampler, as the 'prototype' increment proposal density q! corresponds to a standard normal $N(0,I)$. The increment proposal density can be any other distribution, as long as it is symmetric: it is as likely to produce a random vector $z$ as it is to produce $-z$.

Forming a proposal

Assuming s=RWMState, we can use two methods. The first is draw!, which forms a proposal. This means that starting from $X_{k-1}=$s.x, we produce

\[Y_k \sim X_{k-1} + C U_k, \quad U_k \sim q,\]

where $q$ corresponds to the distribution s.q! simulates from. The increment vector $U_k$ is stored to an internal s.u (which is used by some of the adaptive algorithms), and the proposal $Y_k$ is stored to s.y.

The factor $C$ in the update is not part of the RWMState structure, but supplied as a parameter to the function draw!. It can be a scalar, which controls the increment scale, or a scalar times Cholesky factor, which controls the shape of the increment distribution.

AdaptiveMCMC.draw! — Function

draw!(s::RWMState, sc::AbstractFloat)

Draw proposal from s.x to s.y using scaling sc.

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draw!(s::RWMState, L::Cholesky, sc::AbstractFloat)

Draw proposal from s.x to s.y using scaling sc*L.L, with default sc=1.0.

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Forming a proposal with adaptive state

Calling draw(r::RWMState, s::AdaptState), where s is one of the Adaptation state will use the adapted scale/shape when forming the proposal.

Accepting a proposal

After draw!, the state s.x and the proposal s.y can be accessed (and for instance copied to storage), but should not be modified directly. Instead, the 'acceptance', which means that s.y will replace s.x should be done as follows: