# API for MCMC Proposal Selection

#### Pavel N. Krivitsky

# Summary

This document describes the process by which `ergm` and related packages selects the MCMC proposal for a particular analysis. Note that it is not intended to be a tutorial as much as a description of what inputs and outputs different parts of the system expect. Nor does it cover the C API.

# Description

## Inputs

There is a number of factors that can affect MCMC sampling, some of them historical and some of them new:

Globals

  functions and other structures defined in an accessible namespace

  - `ergm_proposal_table()` a function that if called with no arguments returns a table of registered proposals and updates it otherwise. See `? ergm_proposal_table` for documentation and the meaning of its columns. Of particular interest is its `Constraints` column, which encodes which constraints the proposal does (always) enforce and which it can enforce.
  - `InitErgmReference.<REFERENCE>` a family of initializers for the reference distribution. For the purposes of the proposal selection, among its outputs should be `$name` specifying the name of the reference distribution.
  - `InitErgmConstraint.<CONSTRAINT>` a family of initializers for constraints, weightings, and other high-level specifiers of the proposal distribution. Hard constraints, probabilistic weights, and hints all use this API. For the purposes of the proposal selection, its outputs include
    - `$constrain` (defaulting to `<CONSTRAINT>`) a character vector specifying which constraints are enforced, and can include several semantically nested elements;
    - `$dependence` (defaulting to `TRUE`) specifying whether the constraint is dyad-dependent;
    - `$priority` (defaulting to `Inf`) specifying how important it is that the constraint is met (with `Inf` meaning that it must be met); and
    - `$implies`/`$impliedby` specifying which other constraints this constraint enforces or is enforced by, and this can include itself for constraints, such as `edges` that can only be applied once.
    - `$free_dyads` either an RLEBDM or a function with no arguments that returns an RLEBDM specifying which dyads are not constrained by this constraint.
Arguments

  arguments and settings passed to the call or as control parameters.

  - `constraints=` argument (top-level): A one-sided formula containing a `+`- or `-`-separated list of constraints. `+` terms add additional constraints to the model whereas `-` constraints relax them. `-` constraints are primarily used internally observational process estimation and are not described in detail, except to note that 1) they must be dyad-independent and 2) they necessitate falling back to the RLEBDM sampling API.
  - `reference=` argument (top-level): A one-sided formula specifying the ERGM reference distribution, usually as a name with parameters if appropriate.
  - `control$MCMC.prop=` control parameter: A formula whose RHS containing `+`-separated “hints” to the sampler; an optional LHS may contain the proposal name directly.
  - `control$MCMC.prop.weights=` control parameter: A string selecting proposal weighting (probably deprecated)
  - `control$MCMC.prop.args=` control parameter: A list specifying information to be passed to the proposal

## Code Path

Most of this is implemented in the `ergm_proposal.formula()` method:

1. `InitErgmReference.<REFERENCE>` is called with arguments of `reference=`’s LHS, obtaining the name of the reference.
2. For each term in the following formula’s RHS, the corresponding `InitErgmConstraint.<CONSTRAINT>` function is called and their outputs are stored in a list of initialized constraints (an `ergm_conlist` object). `.dyads` pseudo-constraint is added to dyad-independent constraints (not to hints with `$priority < Inf`).
   1. `constraints=`
   2. `MCMC.prop=`
3. Constraint lists from the previous two steps are concatenated, with redundant constraints removed based on their `$implies`/`$impliedby` settings.
4. Proposal candidates returned by `ergm_proposal_table()` are filtered by `Class`, `Reference`, `Weights` (if `MCMC.prop.weights` differs from `"default"`), and `Proposal` (if the LHS of `MCMC.prop` is provided).
5. Each candidate proposal is “scored” as follows:
   1. If a proposal does enforce a constraint that is not among the requested by the constraints list, it is discarded.
   2. If a proposal cannot enforce a constraint that is among the requested with `priority==Inf`, it is discarded.
   3. For each constraint that is among requested with `priority<Inf` and that the proposal doesn’t and can’t enforce, its (innate, specified in the column of the `ergm_proposal_table()`) `Priority` value is penalised by the `priority` of that constraint.
6. If there are no candidate proposals left, an error is raised.
7. If more than one is left, the proposal with the highest priority (after being penalised for unmet constraints) is selected.