# Discrete probability masses and NA/NaN/Inf in distributions of summary statistics.

### Description

This explains the use of the `boundaries`

attribute of observed statistics withto handle (1) values of the summary statistics that can occur with some probability mass; (2) special values (NA/NaN/Inf) in distributions of summary statistics. This further explains why `Infusion`

handles special values by removing affected distributions unless the `boundaries`

attribute is used.

### Details

Special values may be encountered in an analysis. For example, trying to estimate a regression coefficient when the predictor variable is constant may return a NaN. Since functions such as `refine`

automatically add simulated distributions, this problem must be automatically handled by the user's simulation function or by the package functions, rather than by user's tinkering with the Infusion procedures.

The user must consider what s-he would do if actual data also included NA/NaN/Inf values. If (1) such data would not be used in the statistical analysis, then the simulation procedure must reflect that, otherwise the analysis will be biased. Alternatively (2) if one considers that special values are informative about parameters (in the above example of a regression coefficient, if a constant predictor variable says something about the parameters), then NA/NaN/Inf must be replaced by a numerical value which is flagged to be distinctly handled.

Thus, in case (1) it may be necessary to simulate alternative data until no NaN's are obtained and the target size of the simulated distribution is reached. One solution is for the user to write a simulation function that calls itself recursively until a valid summary statistic is produced. Care is then needed to avoid infinite recursion (which might well indicate unlikely parameter values).

In case (2), it is necessary to assign some (fixed) dummy numerical value to the summary statistics, and to flag this value using the `boundaries`

attribute of the observed summary statistics. The simulation function should return statistic `foo=-1`

(say) instead of `foo=NaN`

, and one should then set `attr(<observed>,"boundaries") <- c(foo=-1)`

.

Without such active decisions by the user, the inference method has no way to determine whether case (1) or (2) holds, and must thus ignore all empirical distributions including NA/NaN/inf. These empirical distributions are thus ignored by the inference functions.

The boundary attribute is also useful to handle all values of the summary statistics that can occur with some probability mass. For example if the estimate `est_p`

of a probability takes values 0 or 1 with positive probability, one should set `attr(<observed>,"boundaries") <- c(p_est=0,p_est=1)`

.