Selecting priors based on measurement error How do you calculate the appropriate prior if you have the measurement error of an instrument? This paragraph is from Cressie's book "Statistics for Spatio-Temporal Data":

It is often the case that some prior information is available
  regarding the measurement-error variance, allowing a fairly
  informative parameter model to be specified. For example, if we are
  assuming conditionally independent measurement errors that are iid
  $Gau(0, \sigma_{\epsilon}^2)$, then we should specify an informative
  prior for $\sigma_{\epsilon}^2$. Say we were interested in ambient air
  temperature, and we saw that the instrument manufacturer’s
  specifications indicated an “error” of $±0.1°C$. Assuming that this
  “error” corresponds to 2 standard deviations (an assumption that
  should be checked!), we might then specify $\sigma_{\epsilon}^{2}$ to
  have a prior mean of $(0.1/2)^2 = 0.0025$. Because of the instrument
  manufacturer’s specification, we would assume a distribution that had
  a clearly defined and fairly narrow peak at 0.0025 (e.g., inverse
  gamma). In fact, we could just fix at 0.0025; however, the data-model
  error may have other components of uncertainty too (Section 7.1). To
  avoid possible identifiability problems with process-model error, it
  is very important that modelers reduce the uncertainty in as much as
  the Science allows, including doing side studies designed to have
  replicated data.

Does anyone know what is the general procedure to obtain the values of a prior as described above (although the paragraph only refers to obtaining the prior mean)?
 A: Two standard methods are


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*Consult the "instrument maker's specifications," as indicated in the quotation.  This is usually a crude fall-back to be used when no other information is available, because (a) what the instrument maker really means by "accuracy" and "precision" is often indeterminate and (b) how the instrument responded when new in a test lab was likely much better than it performs when used in the field.

*Collect replicate samples.  In environmental sampling there are about a half dozen levels at which samples are routinely replicated (and many more at which they could be replicated), with each level used to control for an assignable source of variation.  Such sources may include:


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*Identity of the person taking the sample.

*Preliminary procedures, such as bailing wells, taken before obtaining a sample.

*Variability in the physical sampling process.

*Heterogeneity within the sample volume itself.

*Changes that might occur when preserving and shipping a sample to a laboratory.

*Variations in preliminary laboratory procedures, such as homogenizing a physical sample or digesting it for analysis.

*The identify of the laboratory analyst(s).

*Differences between laboratories.

*Differences between physically distinct instruments, such as two gas chromatographs.

*Drift in instrument calibration over time.

*Diurnal variation.  (This may be natural and systematic but can appear random when sampling times are arbitrary.)



A full quantitative assessment of components of variability can only be obtained by systematically varying each of these factors according to a suitable experimental design.
Usually only the sources believed to contribute the most variability are studied.  For instance, many studies will systematically split a certain portion of the samples once they have been obtained and ship them to two different laboratories.  A study of the differences among the results of those splits can quantify their contribution to the measurement variability.  If enough such splits are obtained, the full distribution of measurement variability can be estimated as a prior in a hierarchical Bayesian spatio-temporal model.  Because many models assuming Gaussian distributions (for each of calculation), obtaining a Gaussian prior eventually comes down to estimating the mean and variance of the differences between the splits.  In more complicated studies, which aim to identify more than one component of variance, this is performed with the Analysis of Variance (ANOVA) apparatus.
One of the benefits of even thinking about these issues is that they help you identify ways to reduce or even eliminate some of these components of error (without ever having to quantify them), thereby getting closer to Cressie & Wikle's ideal of "reducing the uncertainty as much as the science allows."
For an extended worked example (in soil sampling), see
Van Ee, Blume, and Starks, A Rationale for the Assessment of Errors in the Sampling of Soils.  US EPA, May 1990: EPA/600/4-90/013.
