The implicit loss function for errors in soil information

- Decision-making on sample size can be rationalized by the use of loss functions.
- Loss functions for soil information can be hard to define for policy and similar purposes.
- The implicit loss function is the function which makes a particular sample size rational.
- An implicit loss function is computed for information on change of soil carbon stock.

The loss function expresses the costs to an organization that result from decisions made using erroneous information. In closely constrained circumstances, such as remediation of soil on contaminated land prior to development, it has proved possible to compute loss functions and to use these to guide rational decision making on the amount of resource to spend on sampling to collect soil information. In many circumstances it may not be possible to define loss functions prior to decision making on soil sampling. This may be the case when multiple decisions may be based on the soil information and the costs of errors are hard to predict. We propose the implicit loss function as a tool to aid decision making in these circumstances. Conditional on a logistical model which expresses costs of soil sampling as a function of effort, and statistical information from which the error of estimates can be modelled as a function of effort, the implicit loss function is the loss function which makes a particular decision on effort rational. After defining the implicit loss function we compute it for a number of arbitrary decisions on sampling effort for a hypothetical soil monitoring problem. This is based on a logistical model of sampling cost parameterized from a recent survey of soil in County Donegal, Ireland and on statistical parameters estimated with the aid of a process model for change in soil organic carbon. We show how the implicit loss function might provide a basis for reflection on a particular choice of sampling regime, specifically the simple random sample size, by comparing it with the values attributed to soil properties and functions. In a recent study rules were agreed to deal with uncertainty in soil carbon stocks for purposes of carbon trading by treating a percentile of the estimation distribution as the estimated value. We show that this is equivalent to setting a parameter of the implicit loss function, its asymmetry. We then discuss scope for further research to develop and apply the implicit loss function to help decision making by policy makers and regulators.