# Bayesian Inference in Auditing with Partial Prior Information Using Maximum Entropy Priors

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## Abstract

**:**

## 1. Introduction

## 2. The Likelihood Function

**Example**

**1.**

`Mathematica`

^{©}using the command

`NMaximize`.

## 3. The Maximum Entropy Priors

- If $\theta}_{0}=\frac{{\theta}_{L}+{\theta}_{U}}{2$, then $\pi \left(\theta \right)\sim \mathcal{U}({\theta}_{L},{\theta}_{U})$, that is, the uniform distribution on the interval $({\theta}_{L},{\theta}_{U})$.
- If $\theta}_{0}\ne \frac{{\theta}_{L}+{\theta}_{U}}{2$, then$$\pi \left(\theta \right)\propto exp\left\{\lambda \theta \right\},\phantom{\rule{1.em}{0ex}}{\theta}_{L}\le \theta \le {\theta}_{U},$$$$\theta}_{U}+\frac{({\theta}_{L}-{\theta}_{U})exp\{-\lambda {\theta}_{L}\}}{exp\{-\lambda {\theta}_{L}\}-exp\{-\lambda {\theta}_{U}\}}={\theta}_{0$$

## 4. Numerical Illustrations

`Mathematica`

^{©}(version 11.2).

`WinBUGS`code is available as Supplementary Material to this paper). Therefore, both of the new Bayesian upper bounds shown in Table 1 are tighter than the above conventional Bayesian bound. Furthermore, the Bayesian Multinomial–Dirichlet model is fairly sensitive to the dimension of $\mathbf{p}$, a concern which does not arise in the proposed formulation. For instance, the above numerical illustration developed with a non-informative Dirichlet prior over the range 0–100 obtains an unrealistic 95% upper bound of $295,900, in contrast with the MEP upper bound which is $36,000.

## 5. Discussion

## Supplementary Materials

`Mathematica`code:

`Entropy-380586.nb`and

`WinBUGS`code:

`Entropy-380586.odc`

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Theorem**

**A1.**

- (a)
- There exists a nonnegative real–valued function, finite and $\mathcal{B}$-measurable, denoted by ${f}_{\pi}^{\mathcal{B}}\left(\theta \right)$, defined up to a $\pi $-equivalence by$${\int}_{F}{f}_{\pi}^{\mathcal{B}}\left(\theta \right)\pi \left(d\theta \right)={\int}_{F}^{*}f\left(\mathbf{p}\right)\pi \left(d\mathbf{p}\right),\phantom{\rule{1.em}{0ex}}F\in \mathcal{B}.$$
- (b)
- For each $f\left(\mathbf{p}\right)\in {A}_{b}^{+}\left(\mathbf{p}\right),$ there exists a version ${\widehat{f}}_{\pi}^{\mathcal{B}}\left(\theta \right)$ such that:$${\widehat{f}}_{\pi}^{\mathcal{B}}\left(\theta \right)\ge f\left(\mathbf{p}\right)\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}{f}_{\pi}^{\mathcal{B}}\left(\theta \right)\ge f\left(\mathbf{p}\right)\phantom{\rule{0.277778em}{0ex}}a.s.\phantom{\rule{0.277778em}{0ex}}\left[\pi \right]$$
- (c)
- Some basic properties of the above transformation are:
- 1.
- If $f\left(\mathbf{p}\right)\le K,$ for all $\mathbf{p}\in \Pi ,$ then ${f}_{\pi}^{\mathcal{B}}\left(\theta \right)\le K\phantom{\rule{1.em}{0ex}}a.s.\phantom{\rule{0.277778em}{0ex}}\left[\pi \right]$
- 2.
- ${\left(\alpha f\left(\mathbf{p}\right)\right)}_{\pi}^{\mathcal{B}}=\alpha {f}_{\pi}^{\mathcal{B}}\left(\theta \right)\phantom{\rule{1.em}{0ex}}a.s.\phantom{\rule{0.277778em}{0ex}}\left[\pi \right],$ where α is a real constant.
- 3.
- Given two functions $f\left(\mathbf{p}\right),g\left(\mathbf{p}\right),$ then ${(f\left(\mathbf{p}\right)+g\left(\mathbf{p}\right))}_{\pi}^{\mathcal{B}}\le {f}_{\pi}^{\mathcal{B}}\left(\theta \right)+{g}_{\pi}^{\mathcal{B}}\left(\theta \right)\phantom{\rule{1.em}{0ex}}a.s.\phantom{\rule{0.277778em}{0ex}}\left[\pi \right]$.
- 4.
- If the functions $f\left(\mathbf{p}\right),g\left(\mathbf{p}\right)$ satisfy $f\left(\mathbf{p}\right)\ge g\left(\mathbf{p}\right),$ then ${f}_{\pi}^{\mathcal{B}}\left(\theta \right)\ge {g}_{\pi}^{\mathcal{B}}\left(\theta \right)\phantom{\rule{1.em}{0ex}}a.s.\phantom{\rule{0.277778em}{0ex}}\left[\pi \right]$.
- 5.
- ${\mathbb{E}}_{\pi}\left(\right)open="("\; close=")">{f}_{\pi}^{\mathcal{B}}\left(\mathbf{p}\right)$.
- 6.
- If $f\left(\mathbf{p}\right)$ is a $\mathcal{B}$-measurable function, then ${f}_{\pi}^{\mathcal{B}}\left(\theta \right)=f\left(\mathbf{p}\right)\phantom{\rule{1.em}{0ex}}a.s.\phantom{\rule{0.277778em}{0ex}}\left[\pi \right]$.

- (d)
- If $\mathcal{B}$ is generated by a countable partition $\left\{{F}_{n}\right\}$, then for each $f\left(\mathbf{p}\right)\in {A}_{b}^{+}\left(\mathbf{p}\right)$ the step function defined by$${f}^{\mathcal{B}}\left(\mathbf{p}\right)=\sum _{n}\underset{\mathbf{p}\in {F}_{n}}{sup}f\left(\mathbf{p}\right)\xb7{\mathbf{1}}_{{F}_{n}}\left(\mathbf{p}\right)$$

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**Figure 1.**MEP (solid line) and posterior (dashed) distribution of the total amount of error for Auditor #2 in the DUS data example. The posterior distribution associated with the objective uniform prior (Auditor #1) is dotdashed in grey.

**Table 1.**Probabilities of the null hypothesis and 95% posterior quantile of the total error in the test problem.

Prior Information | Pr{H_{0}|DUS Data} | Posterior Quantile θ_{95} |
---|---|---|

Non informative | 0.97 | $44,000 |

MEP | 0.99 | $36,000 |

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## Share and Cite

**MDPI and ACS Style**

Martel-Escobar, M.; Vázquez-Polo, F.-J.; Hernández-Bastida, A.
Bayesian Inference in Auditing with Partial Prior Information Using Maximum Entropy Priors. *Entropy* **2018**, *20*, 919.
https://doi.org/10.3390/e20120919

**AMA Style**

Martel-Escobar M, Vázquez-Polo F-J, Hernández-Bastida A.
Bayesian Inference in Auditing with Partial Prior Information Using Maximum Entropy Priors. *Entropy*. 2018; 20(12):919.
https://doi.org/10.3390/e20120919

**Chicago/Turabian Style**

Martel-Escobar, María, Francisco-José Vázquez-Polo, and Agustín Hernández-Bastida.
2018. "Bayesian Inference in Auditing with Partial Prior Information Using Maximum Entropy Priors" *Entropy* 20, no. 12: 919.
https://doi.org/10.3390/e20120919