### 1. INTRODUCTION

### 2. MATERIALS AND METHODS

### 2.1 Definition of robust Bayesian inference

_{0}and, realizing that any prior “close” to π

_{0}would also be reasonable, choose Γ to consist of all such “close” priors [1, 2]. A rich and attractive class to work with is that of ε-contamination

*ε*determines the amount of probabilistic deviation from the plausible prior

*π*

_{0}that is allowed and 0<

*ε*<1 reflects how “close” is

*π*to

*π*

_{0}.

*Q*is a class of possible contaminations.

*π*varies over Γ. If the variability of the posterior quantity is small, then one can be assured of robustness with respect to the elicitation process. If the variability is large, one does not have robustness with respective to Γ allowing for further investigation or refinement of the contaminated prior.

*Q*= {all distributions} is used, because this

*Q*contains many unreasonable distributions. Indeed, it is argued therein that more reasonable

*Q*are the class of unimodal (U) distributions and the class of all symmetric unimodal (SU) ones. The contaminated prior taking a uniform type will be a pretty sensible prior and the class of uniform type does contain priors with effective tails substantially larger than

*π*

_{0}[2].

### 2.2 Gaussian plume model

*χ*/

_{k}*Q*)

_{s}, for the centerline of the plume at ground level are obtained by setting y and z equal zero and assuming the height of the capping inversion to be much greater than

*H*, the equation for calculating them is given by

### 2.3 Uncertainty in model parameters

### 3. RESULTS AND DISCUSSION

### 3.1 Procedures for analyzing robustness

(1) quantify the plausible priors of the parameters of three inputs, where the plausible priors means to be the information of the priors;

(2) obtain the contaminated priors of the stated parameters;

(3) derive the classes of ε-contamination of all priors by applying the plausible and contaminated priors of the parameters;

(4) calculate the sector-averaged concentrations by applying the classes of ε-contamination of all priors; and

(5) compares the relative errors of medians of the concentrations based on the plausible and contaminated priors.

*μ*is the mean of plausible priors and

*k*is the lower or upper range consistent with the standard deviations of the contaminated priors. By assuming

*ε*as 10%,

*k*was determined to take a more contamination of 10 % than the standard deviations of the plausible ones in this study. From this setting of

*k*, it has been achieved that the classes of Eq. 4 and 5 do contain priors with effective tails substantially larger than

*π*

_{0}. For easy calculation, unimodal contamination of Eq. 5 is divided as two ranges. That is, U

_{1}and U

_{2}contaminations are assumed that their unimodal ones are equal to U[

*μ-k*,

*μ*] and U[

*μ*,

*μ+k*].

### 3.2 Robustness

*μ*, of the plausible prior for protecting a variability of its centering part. The value of lower and upper regions of the unimodal prior was set as [

*μ-k*,

*μ*] or [

*μ*,

*μ+k*] and the unimodal prior was biased to the left or right by centering

*μ*as compared with the symmetric unimodal one. [

*μ-k*,

*μ*] and [

*μ*,

*μ+k*] contaminations are assumed as U

_{1}and U

_{2}contaminations, respectively.

*μ-k*,

*μ*] or [

*μ*,

*μ+k*] was biased to the right or left as compared with that of concentrations reflecting the symmetric unimodal one. It is noted that the parametric values sampled from the unimodal prior are smaller or larger than ones sampled from the symmetric unimodal one due to the lower or upper range of its distribution set as

*μ-k*or

*μ+k*from

*μ*without symmetry. The distribution of concentrations based on the unimodal prior would be, therefore, biased to the right or left as compared with that of concentrations based on the symmetric unimodal one.

*μ-k*,

*μ*] were varied about 1 to 11.1 % for F-class as compared with those of ones using the plausible one, though ε is assumed as 10 % in this study. It was then expected that there was more robust in the results of the concentrations based on the unimodal prior of [

*μ-k*,

*μ*] than the symmetric unimodal one because of decrease of their error ranges. However, the relative errors of the medians using the unimodal prior of [

*μ*,

*μ+k*] were varied about 7.5 to 18.5 % for F-class as compared with those of ones using the plausible one, though ε is assumed as 10 % in this study. It was then expected that the results of the concentrations based on the unimodal prior of [

*μ*,

*μ+k*] was less robust than the symmetric unimodal one and unimodal one of [

*μ-k*,

*μ*].

### 4. CONCLUSION

*ε*-contamination. Though

*ε*was assumed as 10 %, the medians reflecting the symmetric unimodal priors were nearly approximated within 10 % compared with ones reflecting the plausible ones. However, the medians reflecting the unimodal priors were approximated within 20 % for a few downwind distances compared with ones reflecting the plausible ones. From these robust inferences, it is reasonable to apply the symmetric unimodal priors for analyzing the robustness of the Bayesian inferences.