# Bayesian Spatial Split-Population Survival Model with Applications to Democratic Regime Failure and Civil War Recurrence

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## Abstract

**:**

## 1. Introduction

## 2. (Spatial) Split-Population Survival Model

#### 2.1. Model Development

#### 2.2. Markov Chain Monte Carlo Estimation

- Choose a starting point ${\beta}_{0},{\gamma}_{0}$, ${\rho}_{0}$, ${\lambda}_{0}$ and corresponding ${\mathbf{W}}_{0}=\{{W}_{1},\dots ,{W}_{N}\}$ and ${\mathbf{V}}_{0}=\{{V}_{1},\dots ,{V}_{N}\}$, then set $k=0$.
- Update ${\mathsf{\Sigma}}_{\beta}\sim \pi \left({\mathsf{\Sigma}}_{\beta}\right|\beta )$, ${\mathsf{\Sigma}}_{\gamma}\sim \pi \left({\mathsf{\Sigma}}_{\gamma}\right|\gamma )$, $\lambda \sim \pi \left(\lambda \right|\mathbf{W},\mathbf{V})$ using Gibbs sampling. The closed form of the full conditional distributions for $\pi \left({\mathsf{\Sigma}}_{\beta}\right|\beta )$, $\pi \left({\mathsf{\Sigma}}_{\gamma}\right|\gamma )$, $\pi \left(\lambda \right|\mathbf{W},\mathbf{V})$ are derived and defined in the Supplementary Materials.
- Update $\beta \sim \pi \left(\beta \right|\mathbf{C},\mathbf{X},\mathbf{Z},\mathbf{t},\mathbf{W},\mathbf{V},\gamma ,\rho ,{{}^{\u2013}}_{\beta},{\mathsf{\Sigma}}_{\beta})$, $\gamma \sim \pi \left(\gamma \right|\mathbf{C},\mathbf{X},\mathbf{Z},\mathbf{t},\mathbf{W},\mathbf{V},\beta ,\rho ,{{}^{\u2013}}_{\gamma},{\mathsf{\Sigma}}_{\gamma})$, and $\rho \sim \pi \left(\rho \right|\mathbf{C},\mathbf{X},\mathbf{Z},\mathbf{t},\mathbf{W},\mathbf{V},\beta ,\gamma ,{a}_{\rho},{b}_{\rho})$ using the slice sampler with stepout and shrinkage (Neal, 2003); see the Supplementary Materials for details on performing the slice sampling operation in this step.
- Update $\mathbf{W}\sim \pi \left(\mathbf{W}\right|\mathbf{C},\mathbf{X},\mathbf{Z},\mathbf{t},\mathbf{V},\beta ,\gamma ,\rho ,\lambda )$ and $\mathbf{V}\sim \pi \left(\mathbf{V}\right|\mathbf{C},\mathbf{X},\mathbf{Z},\mathbf{t},\mathbf{W},\beta ,\gamma ,\rho ,\lambda )$ via Metropolis–Hastings.
- Set $k=k+1$, then return to Step 2 and repeat for K iterations.

- Choose the initial values of $\beta ,\gamma $, and $\rho $, then set $m=0$.
- Update ${\mathsf{\Sigma}}_{\beta}$ and ${\mathsf{\Sigma}}_{\gamma}$ via Metropolis–Hastings; see the Supplementary Material for the closed form of the full conditional distributions for ${\mathsf{\Sigma}}_{\beta}$ and ${\mathsf{\Sigma}}_{\gamma}$.
- Update $\beta $, $\gamma $, and $\rho $ using the slice sampler with stepout and shrinkage, as described in the Supplementary Materials.
- Repeat Steps 2 and 3 until the chain converges.
- After M iterations, summarize the parameter estimates using posterior samples.

## 3. Monte Carlo Simulations

## 4. Empirical Applications

#### 4.1. Democratic Consolidation and Survival

**A**. We construct a matrix

**A**with elements ${a}_{i{i}^{\prime}}$ such that ${a}_{i{i}^{\prime}}=1$ for each year if the capital of country i is less than 800 km from the capital of country ${i}^{\prime}$ and ${a}_{i{i}^{\prime}}=0$ if countries i and ${i}^{\prime}$ are greater than 800 km from each other. Using geographic proximity as the spatial relationship of interest is appropriate, as it allows the frailties to be correlated with those of neighboring democracies rather than assuming spatial independence even within the same regions. Considering our Bayesian MCMC estimation approach, we incorporate the spatial information in

**A**by employing separate CAR priors for the frailty terms vector

**V**(split-stage) and

**W**(survival-stage), which implies a CAR structure of $\mathbf{V}|\lambda $∼ CAR$\left(\lambda \right)$ and $\mathbf{W}|\lambda $∼ CAR$\left(\lambda \right)$. The spatial SP Weibull model is estimated based on the sample from [17] using the MVN prior and our MCMC algorithm described earlier and assigning the Gamma hyperprior for $\lambda $. Here, we use the hyperparameters $a=1$, $b=1$, ${S}_{\beta}$ = ${I}_{p1}$, ${S}_{\gamma}$ = ${I}_{p2}$, ${\nu}_{\beta}=p1$, ${\nu}_{\gamma}=p2$. Recall that ${\mathsf{\Sigma}}_{\beta}$ is the variance of the MVN prior of the vector $\beta $ for p1-dimensional survival stage covariates and that ${\mathsf{\Sigma}}_{\gamma}$ is the MVN’s prior of the vector $\gamma $ for p2-dimensional split-stage covariates. Hence, when we employ the Inverse Wishart (IW) distribution to estimate both ${\mathsf{\Sigma}}_{\beta}$, in which ${\nu}_{\beta}$ is the hyperparameter, and ${\mathsf{\Sigma}}_{\gamma}$, in which ${\nu}_{\gamma}$ is the hyperparameter, we adopt the values p1 for ${\nu}_{\beta}$ and p2 for ${\nu}_{\gamma}$. Finally, $\lambda \sim \mathrm{Gamma}({a}_{\lambda},{b}_{\lambda})$ with a vague prior $({a}_{\lambda},{b}_{\lambda})=(0.001,1/0.001)$. Our Bayesian SP Weibull model results are based on a set of 50,000 iterations after 4000 burn-in scans and thinning of 10.

#### 4.2. Post-Civil War Peace Duration

**V**) range from −1.19 to 1.49 with a standard deviation of 0.5503, and the survival-stage frailties (

**W**) range from −1.21 to 0.727 with a standard deviation of 0.4498. In both stages, there seem to be regional clusters of frailty values.

## 5. Discussion and Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CAR | Conditionally Autoregressive |

CP | Convergence Probabilities |

d.g.p. | Data Generation Process |

i.i.d. | Independent and Identically Distributed |

IW | Inverse Wishart |

MC | Monte Carlo |

MCMC | Markov Chain Monte Carlo |

MCSE | Monte Carlo Standard Error |

MVN | Multivariate Normal |

NS | Non-Spatial |

NSF | Non-Spatial Frailty |

RMSE | Root Mean Square Error |

SP | Split Population |

## Appendix A

**Figure A1.**Results from the two autocorrelation diagnostics for the democratic survival application: (

**a**) join count and (

**b**) Moran’s I.

**Figure A2.**Results from the two autocorrelation diagnostics for the post-war peace duration application: (

**a**) join count and (

**b**) Moran’s I.

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**Figure 1.**MC Experiment 1 $\beta $,$\gamma $ densities for SP Weibull, NS Frailty Weibull, and Spatial SP Weibull models for: (

**a**) $\widehat{{\beta}_{0}},N=100$, (

**b**) $\widehat{{\beta}_{0}},N=400$, (

**c**) $\widehat{{\beta}_{0}},N=1000$, (

**d**) $\widehat{{\beta}_{0}},N=1500$, (

**e**) $\widehat{{\beta}_{0}},N=2000$, (

**f**) $\widehat{{\beta}_{1}},N=100$, (

**g**) $\widehat{{\beta}_{1}},N=400$, (

**h**) $\widehat{{\beta}_{1}},N=1000$, (

**i**) $\widehat{{\beta}_{1}},N=1500$, (

**j**) $\widehat{{\beta}_{1}},N=2000$, (

**k**) $\widehat{{\gamma}_{0}},N=100$, (

**l**) $\widehat{{\gamma}_{0}},N=400$, (

**m**) $\widehat{{\gamma}_{0}},N=1000$, (

**n**) $\widehat{{\gamma}_{0}},N=1500$, (

**o**) $\widehat{{\gamma}_{0}},N=2000$, (

**p**) $\widehat{{\gamma}_{1}},N=100$, (

**q**) $\widehat{{\gamma}_{1}}$, $N=400$, (

**r**) $\widehat{{\gamma}_{1}},N=1000$, (

**s**) $\widehat{{\gamma}_{1}},N=1500$, (

**t**) $\widehat{{\gamma}_{1}},N=2000$, (

**u**) $\widehat{{\gamma}_{2}},N=100$, (

**v**) $\widehat{{\gamma}_{2}},N=400$, (

**w**) $\widehat{{\gamma}_{2}},N=1000$, (

**x**) $\widehat{{\gamma}_{2}},N=1500$, (

**y**) $\widehat{{\gamma}_{2}},N=2000$.

**Figure 2.**MC Experiment 2 mean RMSE comparison between SP Weibull, NS Frailty SP Weibull, and spatial SP Weibull models for (

**a**) $\widehat{\beta}$ coefficients and (

**b**) $\widehat{\gamma}$ coefficients with spatial dependence changing from 30% to 80% of the data.

**Figure 3.**MC Experiment 3 mean RMSE comparison between SP Weibull, NS Frailty SP Weibull, and spatial SP Weibull models for (

**a**) $\widehat{\beta}$ coefficients and (

**b**) $\widehat{\gamma}$ coefficients with the immune fraction changing from 25% to 60% of the data.

**Figure 4.**Democratic survival application spatial frailty maps: (

**a**) depicts the posterior mean estimates of

**V**(split−stage spatial frailties) and (

**b**) depicts the posterior mean estimates of

**W**(survival−stage spatial frailties).

**Figure 5.**Democratic consolidation stage ($\widehat{\gamma}$) coefficient results from SP Weibull, NS Frailty SP Weibull, and spatial SP Weibull models for the following covariates: (

**a**) GDP/cap, (

**b**) GDP growth, (

**c**) military government, (

**d**) monarchy, (

**e**) civilian government, (

**f**) parliamentary government, and (

**g**) presidential government.

**Figure 6.**Democratic survival stage ($\widehat{\beta}$) coefficient results from SP Weibull, NS Frailty SP Weibull, and Spatial SP Weibull models for the following covariates: (

**a**) GDP/cap, (

**b**) GDP growth, (

**c**) military government, (

**d**) monarchy, (

**e**) civilian government, (

**f**) parliamentary government, and (

**g**) presidential government.

**Figure 7.**Post−war peace duration application spatial frailty maps: (

**a**) depicts the posterior mean estimates of

**V**(split−stage spatial frailties) and (

**b**) depicts the posterior mean estimates of

**W**(survival−stage spatial frailties).

**Figure 8.**Peace consolidation ($\widehat{\gamma}$) coefficient results from SP Weibull and Spatial SP Weibull models for the following covariates: (

**a**) press freedom, (

**b**) victory, (

**c**) mountains, and (

**d**) GDP/cap.

**Figure 9.**Peace survival stage ($\widehat{\beta}$) coefficient results from SP Weibull and Spatial SP Weibull models for the following covariates: (

**a**) press freedom, (

**b**) GDP/cap, (

**c**) peace agreement, (

**d**) intensity, (

**e**) ethnic factionalization, (

**f**) UN peacekeeping, (

**g**) territory, (

**h**) non-contiguous, and (

**i**) mountains.

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**MDPI and ACS Style**

Joo, M.M.; Bolte, B.; Huynh, N.; Mukherjee, B. Bayesian Spatial Split-Population Survival Model with Applications to Democratic Regime Failure and Civil War Recurrence. *Mathematics* **2023**, *11*, 1886.
https://doi.org/10.3390/math11081886

**AMA Style**

Joo MM, Bolte B, Huynh N, Mukherjee B. Bayesian Spatial Split-Population Survival Model with Applications to Democratic Regime Failure and Civil War Recurrence. *Mathematics*. 2023; 11(8):1886.
https://doi.org/10.3390/math11081886

**Chicago/Turabian Style**

Joo, Minnie M., Brandon Bolte, Nguyen Huynh, and Bumba Mukherjee. 2023. "Bayesian Spatial Split-Population Survival Model with Applications to Democratic Regime Failure and Civil War Recurrence" *Mathematics* 11, no. 8: 1886.
https://doi.org/10.3390/math11081886