AR1: Autoregressive Prior of Order 1

In bage: Bayesian Estimation and Forecasting of Age-Specific Rates

Autoregressive Prior of Order 1

Description

Use an autoregressive process of order 1 to model a main effect, or use multiple AR1 processes to model an interaction. Typically used with time effects or with interactions that involve time.

Usage

AR1(
  s = 1,
  shape1 = 5,
  shape2 = 5,
  min = 0.8,
  max = 0.98,
  along = NULL,
  con = c("none", "by")
)

Arguments

s	Scale for the prior for the innovations. Default is 1.
shape1, shape2	Parameters for beta-distribution prior for coefficients. Defaults are 5 and 5.
min, max	Minimum and maximum values for autocorrelation coefficient. Defaults are 0.8 and 0.98.
along	Name of the variable to be used as the 'along' variable. Only used with interactions.
con	Constraints on parameters. Current choices are "none" and "by". Default is "none". See below for details.

Details

If AR() is used with an interaction, separate AR processes are constructed along the 'along' variable, within each combination of the 'by' variables.

Arguments min and max can be used to specify the permissible range for autocorrelation.

Argument s controls the size of innovations. Smaller values for s tend to give smoother estimates.

Value

An object of class "bage_prior_ar".

Mathematical details

When AR1() is used with a main effect,

\beta_j = \phi \beta_{j-1} + \epsilon_j

\epsilon_j \sim \text{N}(0, \omega^2),

and when it is used with an interaction,

\beta_{u,v} = \phi \beta_{u,v-1} + \epsilon_{u,v}

\epsilon_{u,v} \sim \text{N}(0, \omega^2),

where

• \pmb{\beta} is the main effect or interaction;

• j denotes position within the main effect;

• v denotes position within the 'along' variable of the interaction; and

• u denotes position within the 'by' variable(s) of the interaction.

Internally, AR1() derives a value for \omega that gives every element of \beta a marginal variance of \tau^2. Parameter \tau has a half-normal prior

\tau \sim \text{N}^+(0, \mathtt{s}^2),

where s is provided by the user.

Coefficient \phi is constrained to lie between min and max. Its prior distribution is

\phi = (\mathtt{max} - \mathtt{min}) \phi' - \mathtt{min}

where

\phi' \sim \text{Beta}(\mathtt{shape1}, \mathtt{shape2}).

Constraints

With some combinations of terms and priors, the values of the intercept, main effects, and interactions are are only weakly identified. For instance, it may be possible to increase the value of the intercept and reduce the value of the remaining terms in the model with no effect on predicted rates and only a tiny effect on prior probabilities. This weak identifiability is typically harmless. However, in some applications, such as forecasting, or when trying to obtain interpretable values for main effects and interactions, it can be helpful to increase identifiability through the use of constraints.

Current options for constraints are:

• "none" No constraints. The default.

• "by" Only used in interaction terms that include 'along' and 'by' dimensions. Within each value of the 'along' dimension, terms across each 'by' dimension are constrained to sum to 0.

References

• AR1() is based on the TMB function AR1

• The defaults for min and max are based on the defaults for forecast::ets().

See Also

• AR() Generalization of AR1()

• Lin_AR(), Lin_AR1() Line with AR errors

• priors Overview of priors implemented in bage

• set_prior() Specify prior for intercept, main effect, or interaction

Examples

AR1()
AR1(min = 0, max = 1, s = 2.4)
AR1(along = "cohort")

bage documentation built on April 3, 2025, 8:53 p.m.