Proposal name.

Proposal description.

The weight determines how often a Proposal is executed per iteration of the Markov chain.

Proposal dimension.

The number of affected, independent parameters.

The optimal acceptance rate of low dimensional proposals is higher than for high dimensional ones.

Optimal acceptance rates are still subject to controversies. As far as I know, research has focused on random walk proposal with a multivariate normal distribution of dimension d. In this case, the following acceptance rates are desired:

• one dimension: 0.44 (numerical results);
• five and more dimensions: 0.234 (numerical results);
• infinite dimensions: 0.234 (theorem for specific target distributions).

See Handbook of Markov chain Monte Carlo, chapter 4.

Of course, many proposals may not be classical random walk proposals. For example, the beta proposal on a simplex (beta) samples one new variable of the simplex from a beta distribution while rescaling all other variables. What is the dimension of this proposal? I don't know, but I set the dimension to 2. The reason is that if the dimension of the simplex is 2, two variables are changed. If the dimension of the simplex is high, one variable is changed substantially, while all others are changed marginally.

Further, if a proposal changes a number of variables in the same way (and not independently like in a random walk proposal), I still set the dimension of the proposal to the number of variables changed.

Finally, I assume that proposals of unknown dimension have high dimension, and use the optimal acceptance rate 0.234.

A Proposal is an instruction about how the Markov chain will traverse the state space a. Essentially, it is a probability mass or probability density conditioned on the current state (i.e., a Markov kernel).

A Proposal may be tuneable in that it contains information about how to enlarge or shrink the step size to tune the acceptance rate.

Convert a proposal from one data type to another using a lens.

For example:

scaleFirstEntryOfTuple = _1 @~ scale

Simple proposal without tuning information.

Instruction about randomly moving from the current state to a new state, given some source of randomness.

In order to calculate the Metropolis-Hastings-Green ratio, we need to know the ratio of the backward to forward kernels (i.e., the probability masses or probability densities) and the absolute value of the determinant of the Jacobian matrix.

For unbiased proposals, these values are 1.0 such that

proposalSimpleUnbiased x g = return (x', 1.0, 1.0)

For biased proposals, the kernel ratio is qYX / qXY, where qXY is the probability density to move from X to Y, and the absolute value of the determinant of the Jacobian matrix differs from 1.0.

Tune the acceptance rate of a Proposal; see tune, or autoTuneCycle.

Tune the proposal?

Create a tuneable proposal.

Tune a Proposal. Return Nothing if Proposal is not tuneable. The size of the proposal is proportional to the tuning parameter. Negative tuning parameters are not allowed.

See PDimension.

Header of proposal summaries.

Horizontal line of proposal summaries.

Proposal summary.

Define the order in which Proposals are executed in a Cycle. The total number of Proposals per Cycle may differ between Orders (e.g., compare RandomO and RandomReversibleO).

In brief, a Cycle is a list of proposals.

The state of the Markov chain will be logged only after all Proposals in the Cycle have been completed, and the iteration counter will be increased by one. The order in which the Proposals are executed is specified by Order. The default is RandomO.

No proposals with the same name and description are allowed in a Cycle, so that they can be uniquely identified.

Create a Cycle from a list of Proposals.

Set the order of Proposals in a Cycle.

Replicate Proposals according to their weights and possibly shuffle them.

Tune Proposals in the Cycle. See tune.

Calculate acceptance rates and auto tune the Proposals in the Cycle. For now, a Proposal is enlarged when the acceptance rate is above 0.44, and shrunk otherwise. Do not change Proposals that are not tuneable.

Summarize the Proposals in the Cycle. Also report acceptance rates.

For each key k, store the number of accepted and rejected proposals.

In the beginning there was the Word.

Initialize an empty storage of accepted/rejected values.

For key k, prepend an accepted (True) or rejected (False) proposal.

Reset acceptance storage.

Transform keys using the given lists. Keys not provided will not be present in the new Acceptance variable.

Acceptance counts and rate for a specific proposal.

Acceptance rates for all proposals.