# Bayesian Mixture Modelling for Mortality Projection

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## Abstract

**:**

## 1. Introduction

## 2. Bayesian Lee–Carter with CBD

**0**and covariance matrix $\Omega $. The overall model’s density is then specified as below:

## 3. Bayesian Lee–Carter with Age-Cohort

## 4. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**History plot (

**top left**), posterior distribution function (

**top right**), posterior density function (histogram;

**bottom left**), and autocorrelation plot (

**bottom right**) of $\alpha (60)$ (left panel) and ${\kappa}^{(3)}(1970)$ (right panel), with two chains of simulations (pink and blue).

**Figure 2.**Posterior means of $\alpha (x)$, $\beta (x)$, $\kappa (t)$, ${\kappa}^{(1)}(t)$, ${\kappa}^{(2)}(t)$, and ${\kappa}^{(3)}(t)$ from Bayesian Lee–Carter model (alone), CBD model (alone), and mixture model.

**Figure 3.**Standardised residuals by age, calendar year, and cohort year from Bayesian Lee–Carter model (alone), CBD model (alone), and mixture model.

**Figure 4.**Log death rates at ages 60, 70, and 80 from 1970 to 2050 (solid—observed values; dashed—projected values; dotted—95% prediction intervals) under Bayesian Lee–Carter model (alone), CBD model (alone), and mixture model.

**Figure 5.**Posterior means of $\alpha (x)$, $\beta (x)$, $\kappa (t)$, ${\alpha}^{\ast}(x)$, ${\beta}^{\ast}(x)$, and ${\gamma}^{\ast}(c)$ from Bayesian Lee–Carter model (alone), age-cohort model (alone), and mixture model.

**Figure 6.**Standardised residuals by age, calendar year, and cohort year from Bayesian Lee–Carter model (alone), age-cohort model (alone), and mixture model.

**Figure 7.**Log death rates at ages 60, 70, and 80 from 1970 to 2050 (solid—observed values; dashed—projected values; dotted—95% prediction intervals) under Bayesian age-cohort model (alone) and mixture model.

**Figure 8.**Log death rates at ages 60, 70, and 80 from 1970 to 2017 (solid—observed values; dashed—projected values; dotted—95% prediction intervals) under Bayesian age-cohort model (alone) and mixture model.

**Table 1.**Mean absolute errors (MAE) and mean square errors (MSE) of projected log death rates over 2000–2017 under different Bayesian models.

Model | MAE | MSE |
---|---|---|

Lee–Carter | 0.0382 | 0.0022 |

CBD | 0.0462 | 0.0032 |

age-cohort | 0.1284 | 0.0221 |

Lee–Carter + CBD | 0.0439 | 0.0030 |

Lee–Carter + age-cohort | 0.0363 | 0.0020 |

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**MDPI and ACS Style**

Li, J.; Kogure, A.
Bayesian Mixture Modelling for Mortality Projection. *Risks* **2021**, *9*, 76.
https://doi.org/10.3390/risks9040076

**AMA Style**

Li J, Kogure A.
Bayesian Mixture Modelling for Mortality Projection. *Risks*. 2021; 9(4):76.
https://doi.org/10.3390/risks9040076

**Chicago/Turabian Style**

Li, Jackie, and Atsuyuki Kogure.
2021. "Bayesian Mixture Modelling for Mortality Projection" *Risks* 9, no. 4: 76.
https://doi.org/10.3390/risks9040076