I have implemented two models for signal detection theory, namely I defined two likelihoods f_1(y|\theta), and f_2(y|\theta), where y denotes data and \theta a model parameter.

The data y consists of 6 non-netagive valued random variables, i.e., y=(H_1,H_2,H_3,F_1,F_2,F_3) which can be displayed by the following table.

Consider two datasets y_1 and y_2, one of which is obtained as the perturvation of the other.

# data y_1

Confidence Level | Number of True Positives | Number of False Positives |
---|---|---|

3 = definitely present | H_{3}=97 | F_{3}=0 |

2 = equivocal | H_{2}=32 | F_{2}=\color{red}{0} |

1 = questionable | H_{1}=31 | F_{1}=74 |

# data y_2

Confidence Level | Number of True Positives | Number of False Positives |
---|---|---|

3 = definitely present | H_{3}=97 | F_{3}=0 |

2 = equivocal | H_{2}=32 | F_{2}=\color{red}{1} |

1 = questionable | H_{1}=31 | F_{1}=74 |

Because these two datasets are similar, so I expect that the posterior estimates is also similar. But, in one model f_1(y|\theta), the estimates or posterior distribution is unstable and esimates (such as posterior means or posterior variance) are dramtically changed as follows.

# Estimates for data y_1

Param name | mean | se_mean | sd |
---|---|---|---|

z[1] | -5.924416e-01 | 1.185005e-02 | 0.12219574 |

z[2] | 8.760078e+305 | NaN | Inf |

z[3] | 1.860283e+306 | NaN | Inf |

# Estimates for data y_2

Param name | mean | se_mean | sd |
---|---|---|---|

z[1] | -6.197717e-01 | 6.744095e-03 | 1.213770e-01 |

z[2] | 2.440500e+00 | 4.749197e-02 | 5.032560e-01 |

z[3] | 6.120099e+00 | 1.652269e-01 | 1.383218e+00 |

So, I want to quantify this robustness.

So my **question** is :

Is there any famous way to show this robustness?

or any charcteristics for robustness or books?

The codes is open to public, if necesarry, I show it.

To tell the truth, my new model has robustness, so I want to show this robusness as a venefit of my new model.

Is there another question on Bayesian analysis.

In the above fitting, some constant posterior chains appear, then R hat is NaN, but when I visualize the estimates with constant chains, the estimates looks reasonable, because the fitted curve are close to data-points. So, How should we consider these constant chains.

So, another **question** is:

Are fitted estimates not reliable if it includes constant chains?