## Introduction

An understanding of how mortality risk changes with age is needed to anticipate population dynamics (Pollock 1981). Projections of population declines because of climate change, habitat loss and over-exploitation based on assumptions of constant risk or estimates that omit important sources of uncertainty could have disastrous implications for management and conservation. However, studying age-specific mortality (or survival) rates in the wild is challenging because of scarcity of old individuals (Ricklefs & Scheuerlein 2001; Metcalf *et al.* 2009) and unknown times of birth and death for most individuals (Nisbet 2001). Field studies typically span the lifetimes of few individuals, and ages for a large portion of the population are unknown (Frederiksen, Wanless & Harris 2004). All these data constraints stress the need to develop further analytical methods that can account for the sources of uncertainty associated with limited data while extracting the (hidden) demographic processes.

Capture–recapture/recovery (CRR) studies are based on the repeated sampling of a population in which individuals are first marked and released, and, at each subsequent occasion, they are either recaptured, not detected, or recovered dead (Catchpole *et al.* 1998). Observations consist of longitudinal *individual histories* of capture, ideally bounded by times of birth and death (i.e. uncensored records). However, most CRR data sets include individuals with unknown time of birth, some of which could have been born before the study (left-truncated), and individuals with unknown time of death, including (but not restricted to) individuals that died after the termination of the study (right-censored). Models derived from the Cormack–Jolly–Seber framework (CJS; after Cormack 1964; Jolly 1965; Seber 1965) can accommodate both uncensored and right-censored records. This treatment recognizes that both types of observations need inclusion and that each contributes different information (White & Burnham 1999). Often, left-truncated records are included either by assuming that mortality is only time dependent (i.e. constant with age; Aebischer & Coulson 1990) or by using time at first capture as a surrogate for age at maturity (Crespin *et al.* 2006; Reed *et al.* 2008); both treatments bear strong assumptions that can potentially mislead inference on age-specific survival. As a result, in most cases, left-truncated records are discarded. Without left-truncated records, the remaining subset of the population can be small, consisting only of individuals born since the study began. For long-lived species, decades might be required before known-age individuals become a substantial fraction of the population. Yet, deaths may be observed for a variety of individuals, not just those of known age. Although the consequences for inference of ignoring observations on much of the population could be substantial, there is rare opportunity to compare estimates against those that might come from more complete observations.

The missing data problem in CRR data sets does not only pertain to survival status, but also other state variables. CJS-based models provide the basis to estimate survival probabilities for the single-state case when times of death are unknown (i.e. right-censoring), and when recapture probabilities are <1, using likelihoods that marginalize over all the possible states that could apply to each individual. Generalizations of the basic CJS framework have been developed to include multiple states, such as location and developmental stage (Arnason 1973; Schwarz, Schweigert & Arnason 1993; Lebreton & Pradel 2002). The likelihoods necessarily become large and complex to accommodate the many combinations of potential states for each individual, but can be simplified conditionally. Dupuis (1995, 2002) used a Bayesian framework to extend the Arnason–Schwartz model to estimate survival and movement probabilities from capture–recapture data that accounted for missing data on locations and capture occasions through data augmentation. King & Brooks (2002) extended Dupuis (1995) approach when competing models were tested by incorporating model averaging to estimate survival parameters and transition probabilities between states using reversible jump Markov Chain Monte Carlo (RJMCMC) algorithms. Clark *et al.* (2005) developed a hierarchical Bayesian framework that combined Dupuis (1995) approach with stage-structured population modelling (Fujiwara & Caswell 2002) where transitions between states could not be associated with specific covariates and were modelled as random effects. Clark *et al.* (2005) showed that conditional modelling of latent (unknown) states vastly simplifies algorithms thereby allowing for more flexible modelling.

Latent state models have been primarily applied to finite numbers of discrete states (e.g. locations, stages). However, the estimation of initial and terminal states such as times of birth and death needs the integration across continuous variables, without known lower (birth) or upper (death) bounds. Frederiksen, Wanless & Harris (2004) sidestepped this problem by modelling age as a quadratic covariate of survival for black-legged kittiwake data that had a large proportion of unknown ages. The assumptions do not describe survival lacking such a quadratic relationship with age making their methods limited to a small range of species. Zajitschek *et al.* (2009), used life-table analysis originally developed by Müller *et al.* (2004), combined with capture–recapture techniques to estimate age-specific survival rates in male and female black field crickets where times of birth and death were unknown. Link & Barker (2005) and Schofield & Barker (2008) applied a Bayesian approach for open mark–recapture populations that combined estimates of population sizes and per capita birth rates to model missing birth times as the multinomial probability of being born in any given interval within the study span. Recently, Pledger *et al.* (2009) developed an extension of the Jolly–Seber and Arnason–Schwartz models to estimate times of arrival and departure at a stopover site, analogous to the estimation of times of birth and death in the previous example. All these approaches (except for Frederiksen, Wanless & Harris 2004) require that the times for the initial and terminal states are approximately known, as with stopover sites or species for which birth and death happen within the same year (e.g. semelparous species). In such cases, it is easier to assume that the population has reached a stable age distribution to approximate survival rates (Zajitschek *et al.* 2009), or to use a maximum-likelihood framework to convolve over all possible initial and terminal states (Link & Barker 2005; Schofield & Barker 2008; Pledger *et al.* 2009). However, in populations consisting of a large number of cohorts with overlapping generations, the marginal likelihood for the estimation of times of birth and death would not be available (see Matechou 2010 for alternatives based on Pledger *et al.* 2009).

Here, we developed an alternative approach that combines estimation of survival parameters and imputation of unknown states (i.e. times of birth and death) within a Bayesian hierarchical framework. The modelling of missing data allowed us to combine what is known from a large number of partially observed individuals to obtain population-level estimates of survival. By modelling both (past) birth years and (future and unknown) death years as latent variables, combined with a flexible parametric mortality function for the full population, we could readily admit partial observations on individuals of unknown age, extending the types of observations that could be included. We simulated data sets of varying duration, detection and recovery probabilities and proportions of individuals with known time of birth. Under these scenarios, we compared our model to a traditional CRR model (Catchpole *et al.* 1998) and assessed both models on the basis of their ability to predict age-specific survival trends and, for our model, parameter estimates and estimates of ages at death. To illustrate the performance of our model, we applied it to a Soay sheep (*Ovis aries*, L.) CRR data set and extended it to accommodate covariates (i.e. sex). Our results showed that within a hierarchical setting, the assimilation of types of observations that are not typically accommodated by a traditional approach could provide more powerful inference than traditional methods.