Mcmc.Proposal.Hamiltonian.Common

Creation date: Fri Jun 3 10:27:12 2022.

See Mcmc.Proposal.Hamiltonian.Hamiltonian.

References:

• [1] Chapter 5 of Handbook of Monte Carlo: Neal, R. M., MCMC Using Hamiltonian Dynamics, In S. Brooks, A. Gelman, G. Jones, & X. Meng (Eds.), Handbook of Markov Chain Monte Carlo (2011), CRC press.
• [2] Gelman, A., Carlin, J. B., Stern, H. S., & Rubin, D. B., Bayesian data analysis (2014), CRC Press.
• [3] Review by Betancourt and notes: Betancourt, M., A conceptual introduction to Hamiltonian Monte Carlo, arXiv, 1701–02434 (2017).
• [4] Matthew D. Hoffman, Andrew Gelman (2014) The No-U-Turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo, Journal of Machine Learning Research.

Basic types

type Positions = Vector Double

The Hamiltonian proposal acts on a vector of floating point values referred to as positions.

The positions can represent the complete state or a subset of the state of the Markov chain; see HStructure.

The length of the position vector determines the size of the squared mass matrix; see Masses.

type Momenta = Vector Double

Momenta of the Positions.

type Masses = Herm Double

Mass matrix.

The masses roughly describe how reluctant the particles move through the state space. If a parameter has higher mass the velocity in this direction will be changed less by the provided gradient than when the same parameter has lower mass. Off-diagonal entries describe the covariance structure. If two parameters are negatively correlated, their generated initial momenta are likely to have opposite signs.

The matrix is square with the number of rows and columns being equal to the length of Positions.

The proposal is more efficient if masses are assigned according to the inverse (co)-variance structure of the posterior function. That is, parameters changing on larger scales should have lower masses than parameters changing on lower scales. In particular, the optimal entries of the diagonal of the mass matrix are the inverted variances of the parameters distributed according to the posterior function.

Of course, the scales of the parameters of the posterior function are usually unknown. Often, it is sufficient to

• set the diagonal entries of the mass matrix to identical values roughly scaled with the inverted estimated average variance of the posterior function; or even to
• set all diagonal entries of the mass matrix to 1.0, and all other entries to 0.0, and trust the tuning algorithm (see HTuningConf) to find the correct values.

type LeapfrogTrajectoryLength = Int

Mean leapfrog trajectory length \(L\).

Number of leapfrog steps per proposal.

To avoid problems with ergodicity, the actual number of leapfrog steps is sampled per proposal from a discrete uniform distribution over the interval \([\text{floor}(0.9L),\text{ceiling}(1.1L)]\).

For a discussion of ergodicity and reasons why randomization is important, see [1] p. 15; also mentioned in [2] p. 304.

To avoid errors, the left bound of the interval has an additional hard minimum of 1, and the right bound is required to be larger equal than the left bound.

type LeapfrogScalingFactor = Double

Mean of leapfrog scaling factor \(\epsilon\).

Determines the size of each leapfrog step.

To avoid problems with ergodicity, the actual leapfrog scaling factor is sampled per proposal from a continuous uniform distribution over the interval \((0.9\epsilon,1.1\epsilon]\).

For a discussion of ergodicity and reasons why randomization is important, see [1] p. 15; also mentioned in [2] p. 304.

Call error if value is zero or negative.

type LeapfrogSimulationLength = Double

Product of LeapfrogTrajectoryLength and LeapfrogScalingFactor.

A good value is hard to find and varies between applications. For example, see Figure 6 in [4].

Call error if value is zero or negative.

Tuning

data HTuneLeapfrog

Tune leapfrog parameters?

Constructors

HNoTuneLeapfrog

HTuneLeapfrog

We expect that the larger the leapfrog scaling factor the lower the acceptance ratio. Consequently, if the acceptance rate is too low, the leapfrog scaling factor is decreased and vice versa. Further, the leapfrog trajectory length is scaled such that the product of the leapfrog scaling factor and leapfrog trajectory length stays constant.

Instances

Show HTuneLeapfrog

Defined in Mcmc.Proposal.Hamiltonian.Common

Methods

showsPrec :: Int -> HTuneLeapfrog -> ShowS

show :: HTuneLeapfrog -> String

showList :: [HTuneLeapfrog] -> ShowS

Eq HTuneLeapfrog

Defined in Mcmc.Proposal.Hamiltonian.Common

Methods

(==) :: HTuneLeapfrog -> HTuneLeapfrog -> Bool

(/=) :: HTuneLeapfrog -> HTuneLeapfrog -> Bool

data HTuneMasses

Tune masses?

The masses are tuned according to the (co)variances of the parameters obtained from the posterior distribution over the last auto tuning interval.

Constructors

HNoTuneMasses

HTuneDiagonalMassesOnly

Diagonal only: The variances of the parameters are calculated and the masses are amended using the old masses and the inverted variances. If, for a specific coordinate, the sample size is 60 or lower, or if the calculated variance is out of predefined bounds [1e-8, 1e8], the mass of the affected position is not changed.

HTuneAllMasses

All masses: The covariance matrix of the parameters is estimated and the inverted matrix (sometimes called precision matrix) is used as mass matrix. This procedure is error prone, but models with high correlations between parameters strongly profit from tuning off-diagonal entries. The full mass matrix is only tuned if equal or more than "(n_masses + max(n_masses, 61)" samples are available. For these reasons, when tuning all masses it is recommended to use tuning settings such as

BurnInWithCustomAutoTuning [10, 20 .. 200] [200, 220 .. 500]

Instances

Show HTuneMasses

Defined in Mcmc.Proposal.Hamiltonian.Common

Methods

showsPrec :: Int -> HTuneMasses -> ShowS

show :: HTuneMasses -> String

showList :: [HTuneMasses] -> ShowS

Eq HTuneMasses

Defined in Mcmc.Proposal.Hamiltonian.Common

Methods

(==) :: HTuneMasses -> HTuneMasses -> Bool

(/=) :: HTuneMasses -> HTuneMasses -> Bool

data HTuningConf

Tuning configuration of the Hamilton Monte Carlo proposal.

Constructors

HTuningConf HTuneLeapfrog HTuneMasses

Instances

Show HTuningConf

Defined in Mcmc.Proposal.Hamiltonian.Common

Methods

showsPrec :: Int -> HTuningConf -> ShowS

show :: HTuningConf -> String

showList :: [HTuningConf] -> ShowS

Eq HTuningConf

Defined in Mcmc.Proposal.Hamiltonian.Common

Methods

(==) :: HTuningConf -> HTuningConf -> Bool

(/=) :: HTuningConf -> HTuningConf -> Bool

Structure of state

data HStructure s

The Hamilton Monte Carlo proposal requires information about the structure of the state, which is denoted as s.

Please also refer to the top level module documentation of Mcmc.Proposal.Hamiltonian.Hamiltonian.

Constructors

HStructure

Fields

• hSample :: s Double

  The sample state is used for error checks and to calculate the dimension of the proposal.

• hToVector :: s Double -> Positions

  Extract a a subset of values to be manipulated by the Hamiltonian proposal from the complete state.

• hFromVectorWith :: s Double -> Positions -> s Double

  Put those values back into the complete state.

Target function

data HTarget s

The target is composed of the prior, likelihood, and jacobian functions.

The structure of the state is denoted as s.

Please also refer to the top level module documentation.

Constructors

HTarget

Fields

• hPrior :: forall a. (RealFloat a, Typeable a) => Maybe (PriorFunctionG (s a) a)

  Function computing the log prior.

• hLikelihood :: forall a. (RealFloat a, Typeable a) => LikelihoodFunctionG (s a) a

  Function computing the log likelihood.

• hJacobian :: forall a. (RealFloat a, Typeable a) => Maybe (JacobianFunctionG (s a) a)

  Function computing the log of the Jacobian.