Elicit priors and initialization from background dataset

Here we fix the hyperparameters for priors on $\theta_i$ and $W_i$ , i.e., $B$ , $U$ , $\mu$ and $\nu_w$ .

make_priors_and_init(
  df.background,
  col.variables,
  col.item,
  use.priors = c("ML", "vague"),
  use.init = c("random", "vague"),
  ...
)

Arguments

df.background	the background dataset
col.variables	columns with variables: names or positions
col.item	column with the id of the item ( $i$ ): names or positions
use.priors	see details
use.init	see details
...	additional variables for priors and init (see the description)

Value

a list of variables

Details

An appropriate initialization value for $W^{-1}_i$ , $i=1,2$ ($h_p$ and $h_d$ chains respectively) is also generated.

Notice that we have three chains in the LR computations, the same initialization is used thrice.

Priors

use.priors:

'ML': maximum likelihood estimation as in (Bozza et al. 2008)
'vague': low-information priors

U is alpha*diag(p), B is beta*diag(p), mu is mu0.

By default alpha = 1, beta = 1, mu0 = 0

The Wishart dofs nw are set as small as possible without losing full rank: $\nu_w = 2(p + 1) + 1$

Initialization

Generate values for $W^{-1}_1$ , $W^{-1}_2$ that will be used as starting values for the Gibbs chains. Two methods:

use.init:

'random': initialize according to the model
- generate one $W_i \sim IW_p(\nu_w, U)$
- then invert: $W^{-1}_i = (W_i)^{-1}$
'vague': low-information initialization
- $W_i = \alpha_{\text{init}} + \beta_{\text{init}} * \operatorname{diag}(p)$

Parameters

Some constants can be changed by passing the new values to ... :

use.priors = 'vague': accepts parameters
- alpha (default: 1)
- beta (default: 1)
- mu0 (default: 0)
use.init = 'vague': accepts parameters
- alpha_init (default: 1)
- beta_init (default: 100)

Return values

A list of variables:

mu: the global mean
B.inv: the between-source prior covariance matrix, as the inverse
U: the Inverted Wishart scale matrix
nw: the Inverted Wishart dof $\nu_w$
W.inv.1, W.inv.2: the initializations for the within-source covariance matrices, as their inverses

Examples

# Use the iris data
head(iris, 3)
#>   Sepal.Length Sepal.Width Petal.Length Petal.Width Species
#> 1          5.1         3.5          1.4         0.2  setosa
#> 2          4.9         3.0          1.4         0.2  setosa
#> 3          4.7         3.2          1.3         0.2  setosa
col_variables <- c(1,3)
col_item <- 5

# Elicitation using MLE
priors_init <- make_priors_and_init(
   df.background = iris,
   col.variables = col_variables,
   col.item = col_item
)

priors_init
#> $mu
#>                  [,1]
#> Sepal.Length 5.843333
#> Petal.Length 3.758000
#> 
#> $U
#>              Sepal.Length Petal.Length
#> Sepal.Length    0.2650082    0.1675143
#> Petal.Length    0.1675143    0.1851878
#> 
#> $B.inv
#>           [,1]      [,2]
#> [1,] 135.25611 -51.13409
#> [2,] -51.13409  19.56022
#> 
#> $nw
#> [1] 7
#> 
#> $W.inv.1
#>           [,1]      [,2]
#> [1,]  4.852533 -4.417948
#> [2,] -4.417948  6.689959
#> 
#> $W.inv.2
#>           [,1]      [,2]
#> [1,]  4.852533 -4.417948
#> [2,] -4.417948  6.689959
#> 

priors_init_2 <- make_priors_and_init(
   df.background = iris,
   col.variables = col_variables,
   col.item = col_item,
   use.priors = "vague",
   alpha_init = 100
)

priors_init_2
#> $mu
#> [1] 0 0
#> 
#> $U
#>      [,1] [,2]
#> [1,]    1    0
#> [2,]    0    1
#> 
#> $B.inv
#>      [,1] [,2]
#> [1,]    1    0
#> [2,]    0    1
#> 
#> $nw
#> [1] 7
#> 
#> $W.inv.1
#>           [,1]      [,2]
#> [1,]  4.715651 -3.956177
#> [2,] -3.956177  8.275267
#> 
#> $W.inv.2
#>           [,1]      [,2]
#> [1,]  4.715651 -3.956177
#> [2,] -3.956177  8.275267
#>