# Evidence and Credibility: Full Bayesian Significance Test for Precise Hypotheses

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Evidence Calculus

^{*}as the maximum of the posterior density over the null hypothesis, attained at the argument $\theta *$,

^{*}for Examples 2 and 3 of Section 4.

^{*}. That is, the evidence of the null hypothesis is

^{*}is “large”, it means that the null set is in a region of low probability and the evidence in the data is against the null hypothesis. On the other hand, if the probability of T

^{*}is “small”, then the null set is in a region of high probability and the evidence in the data is in favor of the null hypothesis.

## 3. Numerical Computation

^{n}, and the hypothesis is defined as a further restricted subset ${\Theta}_{0}\subset \Theta \subseteq {R}^{n}$. Usually, ${\Theta}_{0}$is defined by vector valued inequality and equality constraints:

- Numerical Optimization step:$$\theta *\in \mathrm{arg}\underset{\theta \in {\Theta}_{0}}{\mathrm{max}}f(\theta ),\varphi =f*=f(\theta *)$$
- Numerical Integration step:

^{*}and to calculate $\kappa *$ can be used under general conditions. Our purpose, however, is to discuss precise hypothesis testing, under absolute continuity of the posterior probability model, the case for which most solutions presented in the literature are controversial.

## 4. Examples

- Our measure of evidence, Ev, for each d;
- the p-value, pV obtained by the ${\chi}^{2}$ test; that is, the tail area;
- the Bayes Factor,$$BF=\frac{\mathrm{Pr}\left\{{\Theta}_{0}\right\}\mathrm{Pr}\left\{d|{\Theta}_{0}\right\}}{\left(1-\mathrm{Pr}\left\{{\Theta}_{0}\right\}\right)\mathrm{Pr}\left\{d|\Theta -{\Theta}_{0}\right\}};\mathrm{and}$$
- the posterior probability of H,$$PP=\mathrm{Pr}\left\{{\Theta}_{0}|d\right\}={\left\{\text{\hspace{0.17em}}1+{(BF)}^{-1}\right\}}^{-1}.$$

#### 4.1. Success rate in standard binomial model

d | Ev | PV | BF | PP |
---|---|---|---|---|

0 | 0.00 | 0.00 | 0.00 | 0.00 |

1 | 0.00 | 0.00 | 0.00 | 0.00 |

2 | 0.00 | 0.00 | 0.00 | 0.00 |

3 | 0.00 | 0.00 | 0.02 | 0.02 |

4 | 0.01 | 0.01 | 0.10 | 0.09 |

5 | 0.02 | 0.03 | 0.31 | 0.24 |

6 | 0.06 | 0.07 | 0.78 | 0.44 |

7 | 0.16 | 0.18 | 1.55 | 0.61 |

8 | 0.35 | 0.37 | 2.52 | 0.72 |

9 | 0.64 | 0.65 | 3.36 | 0.77 |

10 | 1.00 | 1.00 | 3.70 | 0.79 |

#### 4.2. Homogeneity test in 2× 2 contingency table

Homogeneity | Hardy-Weinberg | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

x | y | Ev | pV | BF | PP | x_{1} | x_{3} | Ev | pV | BF | PP |

5 | 0 | 0.05 | 0.02 | 0.25 | 0.20 | 1 | 2 | 0.01 | 0.00 | 0.01 | 0.01 |

5 | 1 | 0.18 | 0.08 | 0.87 | 0.46 | 1 | 3 | 0.01 | 0.01 | 0.04 | 0.04 |

5 | 2 | 0.43 | 0.21 | 1.70 | 0.63 | 1 | 4 | 0.04 | 0.02 | 0.11 | 0.10 |

5 | 3 | 0.71 | 0.43 | 2.47 | 0.71 | 1 | 5 | 0.09 | 0.04 | 0.25 | 0.20 |

5 | 4 | 0.93 | 0.71 | 2.95 | 0.75 | 1 | 6 | 0.18 | 0.08 | 0.46 | 0.32 |

5 | 5 | 1.00 | 1.00 | 3.05 | 0.75 | 1 | 7 | 0.31 | 0.15 | 0.77 | 0.44 |

5 | 6 | 0.94 | 0.72 | 2.80 | 0.74 | 1 | 8 | 0.48 | 0.26 | 1.16 | 0.54 |

5 | 7 | 0.77 | 0.49 | 2.31 | 0.70 | 1 | 9 | 0.66 | 0.39 | 1.59 | 0.61 |

5 | 8 | 0.58 | 0.31 | 1.75 | 0.64 | 1 | 10 | 0.83 | 0.57 | 2.00 | 0.67 |

5 | 9 | 0.39 | 0.18 | 1.21 | 0.55 | 1 | 11 | 0.95 | 0.77 | 2.34 | 0.70 |

5 | 10 | 0.24 | 0.10 | 0.77 | 0.43 | 1 | 12 | 1.00 | 0.99 | 2.55 | 0.72 |

10 | 0 | 0.00 | 0.00 | 0.00 | 0.00 | 1 | 13 | 0.96 | 0.78 | 2.57 | 0.72 |

10 | 1 | 0.00 | 0.00 | 0.02 | 0.02 | 1 | 14 | 0.84 | 0.55 | 2.39 | 0.71 |

10 | 2 | 0.01 | 0.01 | 0.07 | 0.06 | 1 | 15 | 0.66 | 0.33 | 2.05 | 0.67 |

10 | 3 | 0.05 | 0.02 | 0.19 | 0.16 | 1 | 16 | 0.47 | 0.16 | 1.58 | 0.61 |

10 | 4 | 0.12 | 0.05 | 0.41 | 0.29 | 1 | 17 | 0.27 | 0.05 | 1.06 | 0.51 |

10 | 5 | 0.24 | 0.10 | 0.77 | 0.43 | 1 | 18 | 0.12 | 0.00 | 0.58 | 0.37 |

10 | 6 | 0.41 | 0.20 | 1.23 | 0.55 | 5 | 0 | 0.02 | 0.01 | 0.05 | 0.05 |

10 | 7 | 0.61 | 0.34 | 1.74 | 0.63 | 5 | 1 | 0.09 | 0.04 | 0.25 | 0.20 |

10 | 8 | 0.81 | 0.53 | 2.21 | 0.69 | 5 | 2 | 0.29 | 0.14 | 0.60 | 0.38 |

10 | 9 | 0.95 | 0.75 | 2.54 | 0.72 | 5 | 3 | 0.61 | 0.34 | 1.00 | 0.50 |

10 | 10 | 1.00 | 1.00 | 2.66 | 0.73 | 5 | 4 | 0.89 | 0.65 | 1.29 | 0.56 |

12 | 0 | 0.00 | 0.00 | 0.00 | 0.00 | 5 | 5 | 1.00 | 1.00 | 1.34 | 0.57 |

12 | 1 | 0.00 | 0.00 | 0.00 | 0.00 | 5 | 6 | 0.90 | 0.66 | 1.18 | 0.54 |

12 | 2 | 0.00 | 0.00 | 0.01 | 0.01 | 5 | 7 | 0.66 | 0.39 | 0.89 | 0.47 |

12 | 3 | 0.01 | 0.00 | 0.04 | 0.04 | 5 | 8 | 0.40 | 0.20 | 0.58 | 0.37 |

12 | 4 | 0.03 | 0.01 | 0.10 | 0.09 | 5 | 9 | 0.21 | 0.09 | 0.32 | 0.24 |

12 | 5 | 0.07 | 0.03 | 0.24 | 0.19 | 5 | 10 | 0.09 | 0.04 | 0.16 | 0.13 |

12 | 6 | 0.14 | 0.06 | 0.46 | 0.32 | 9 | 0 | 0.21 | 0.09 | 0.73 | 0.42 |

12 | 7 | 0.26 | 0.11 | 0.80 | 0.44 | 9 | 1 | 0.66 | 0.39 | 1.59 | 0.61 |

12 | 8 | 0.42 | 0.21 | 1.24 | 0.55 | 9 | 2 | 0.99 | 0.91 | 1.77 | 0.64 |

12 | 9 | 0.62 | 0.34 | 1.73 | 0.63 | 9 | 3 | 0.86 | 0.59 | 1.33 | 0.57 |

12 | 10 | 0.81 | 0.53 | 2.21 | 0.69 | 9 | 4 | 0.49 | 0.26 | 0.74 | 0.43 |

9 | 5 | 0.21 | 0.09 | 0.32 | 0.24 | ||||||

9 | 6 | 0.06 | 0.03 | 0.11 | 0.10 | ||||||

9 | 7 | 0.01 | 0.01 | 0.03 | 0.03 |

#### 4.3. Hardy-Weinberg equilibrium law

_{1}and x

_{3}are the two homozigote sample counts and ${x}_{2}=n-{x}_{1}-{x}_{3}$ is hetherozigote sample count. $\theta =\left[{\theta}_{1},{\theta}_{2},{\theta}_{3}\right]$ is the parameter vector. The posterior density for this trinomial model is

#### 4.4. Independence test in a 2× 2 contingency table

x_{00} | x_{01} | x_{10} | x_{11} | Ev | pV | BF | PP |
---|---|---|---|---|---|---|---|

12 | 6 | 95 | 35 | 0.96 | 0.57 | 4.73 | 0.83 |

48 | 25 | 9 | 10 | 0.54 | 0.14 | 1.04 | 0.51 |

96 | 50 | 18 | 20 | 0.24 | 0.04 | 0.50 | 0.33 |

18 | 5 | 39 | 30 | 0.29 | 0.06 | 0.50 | 0.33 |

36 | 10 | 78 | 60 | 0.06 | 0.01 | 0.11 | 0.10 |

#### 4.5. Comparison of two gamma distributions

Brand 1 sample | ||||

39.27 | 31.72 | 12.33 | 27.67 | 56.66 |

28.32 | 53.72 | 29.71 | 23.76 | 33.55 |

mean_{1}=33.67 | std_{1}=13.33 | |||

Brand 2 sample | ||||

28.32 | 53.72 | 29.71 | 23.76 | 33.55 |

24.07 | 33.79 | 33.10 | 26.93 | 27.23 |

mean_{2}=29.25 | std_{2}=3.62 | |||

Evidence | ||||

$Ev(H\text{'})=0.89$ | $Ev(H)=0.01$ |

## 5. Final Remarks

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**MDPI and ACS Style**

De Bragança Pereira, C.A.; Stern, J.M.
Evidence and Credibility: Full Bayesian Significance Test for Precise Hypotheses. *Entropy* **1999**, *1*, 99-110.
https://doi.org/10.3390/e1040099

**AMA Style**

De Bragança Pereira CA, Stern JM.
Evidence and Credibility: Full Bayesian Significance Test for Precise Hypotheses. *Entropy*. 1999; 1(4):99-110.
https://doi.org/10.3390/e1040099

**Chicago/Turabian Style**

De Bragança Pereira, Carlos Alberto, and Julio Michael Stern.
1999. "Evidence and Credibility: Full Bayesian Significance Test for Precise Hypotheses" *Entropy* 1, no. 4: 99-110.
https://doi.org/10.3390/e1040099