Population projection matrices are a common means for predicting short- and long-term population persistence for rare, threatened and endangered species. Data from such species can suffer from small sample sizes and consequently miss rare demographic events resulting in incomplete or biologically unrealistic life cycle trajectories. Matrices with missing values (zeros; e.g., no observation of seeds transitioning to seedlings) are often patched using prior information from the literature, other populations, time periods, other species, best guess estimates, or are sometimes even ignored. To alleviate this problem, we propose using a multinomial-Dirichlet model for parameterizing transitions and a Gamma for reproduction to patch missing values in these holey matrices. This formally integrates prior information within a Bayesian framework and explicitly includes the weight of the prior information on the posterior distributions. We show using two real data sets that the weight assigned to the prior information mainly influences the dispersion of the posteriors, the inclusion of priors results in irreducible and ergodic matrices, and more biologically realistic inferences can be made on the transition probabilities. Because the priors are explicitly stated, the results are reproducible and can be re-evaluated if alternative priors are available in the future.

种群预测矩阵是预测稀有，受威胁和濒危物种的短期和长期种群持久性的一种常用方法。来自此类物种的数据可能会遭受小样本量的困扰，因此会错过罕见的人口统计事件，从而导致生命周期轨迹不完整或生物学上不切实际。缺少值的矩阵（零；例如，没有观察到种子过渡到幼苗）通常使用来自文献，其他种群，时间段，其他物种，最佳猜测估计的先验信息进行修补，或者有时甚至被忽略。为了缓解此问题，我们建议使用多项式Dirichlet模型对过渡进行参数化，并使用Gamma进行再现以修补这些有孔矩阵中的缺失值。这正式地将先验信息整合在贝叶斯框架内，并明确包括后验分布上先验信息的权重。我们使用两个真实的数据集显示，分配给先验信息的权重主要影响后验者的分散，先验信息的包含导致不可约和遍历矩阵的产生，并且可以对跃迁概率做出更生物学的现实推断。由于先验已明确说明，因此，如果将来可以使用替代先验，则结果是可重现的，并且可以重新评估。先验的包含会导致不可约和遍历矩阵，并且可以从生物学上更真实地推断出转移概率。由于先验已明确说明，因此，如果将来可以使用替代先验，则结果是可重现的，并且可以重新评估。先验的包含会导致不可约和遍历矩阵，并且可以从生物学上更真实地推断出转移概率。由于先验已明确说明，因此，如果将来可以使用替代先验，则结果是可重现的，并且可以重新评估。