# Bayesian Testing of a Point Null Hypothesis Based on the Latent Information Prior

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Kullback-Leibler Loss of Predictive Densities

**Lemma 1.**

## 3. Latent Information Priors

**Theorem 1.**

**Lemma 2.**

**Figure 2.**Risk functions of Bayesian testing based on latent information priors for (

**a**) $w=0.5$ and for (

**b**) $w=0.355$. When $w=0.5$, $a=1.21$ and $b=7$. When $w=0.355$, $a=1.10$ and $b=7$. The vertical dotted lines indicate the locations of a and b.

**Figure 3.**Posterior probabilities ${p}_{w,{\pi}^{*}}(m=0\mid x)$ based on latent information priors for (

**a**) $w=0.5$ and for (

**b**) $w=0.355$.

x | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|

${p}_{w=0.5}(m=0\mid x)$ | 0.702 | 0.564 | 0.295 | 0.112 | 0.0217 |

${p}_{w=0.355}(m=0\mid x)$ | 0.560 | 0.434 | 0.220 | 0.0867 | 0.0145 |

p-value (two-sided test) | 1 | 0.317 | 0.0455 | 0.00267 | $6.33\times {10}^{-5}$ |

## 4. Other Common Priors

#### 4.1. The Normal Prior

**Figure 4.**Risk functions of Bayesian testing based on normal priors for (

**a**) $w=0.5$ and $\tau =4.92$; and for (

**b**) $w=0.355$ and $\tau =5.36$. The functions have symmetry ${r}_{w}(-\mu ,{\mathrm{N}}_{\tau})={r}_{w}(\mu ,{\mathrm{N}}_{\tau})$ about the origin.

#### 4.2. The Cauchy Prior

**Figure 5.**Risk functions of Bayesian testing based on Cauchy priors for (

**a**) $w=0.5$ and $\gamma =3.31$; and for (

**b**) $w=0.355$ and $\gamma =3.63$. The functions have symmetry ${r}_{w}(-\mu ,{\mathrm{C}}_{\gamma})={r}_{w}(\mu ,{\mathrm{C}}_{\gamma})$ about the origin.

## 5. Conclusions

## Acknowledgments

## Conflicts of Interest

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## Appendix. Proofs of Lemmas

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Komaki, F.
Bayesian Testing of a Point Null Hypothesis Based on the Latent Information Prior. *Entropy* **2013**, *15*, 4416-4431.
https://doi.org/10.3390/e15104416

**AMA Style**

Komaki F.
Bayesian Testing of a Point Null Hypothesis Based on the Latent Information Prior. *Entropy*. 2013; 15(10):4416-4431.
https://doi.org/10.3390/e15104416

**Chicago/Turabian Style**

Komaki, Fumiyasu.
2013. "Bayesian Testing of a Point Null Hypothesis Based on the Latent Information Prior" *Entropy* 15, no. 10: 4416-4431.
https://doi.org/10.3390/e15104416