State-space models are commonly used to describe different forms of ecological data. We consider the case of count data with observation errors. For such data the system process is typically multi-dimensional consisting of coupled Markov processes, where each component corresponds to a different characterisation of the population, such as age group, gender or breeding status. The associated system process equations describe the biological mechanisms under which the system evolves over time. However, there is often limited information in the count data alone to sensibly estimate demographic parameters of interest, so these are often combined with additional ecological observations leading to an integrated data analysis. Unfortunately, fitting these models to the data can be challenging, especially if the state-space model for the count data is non-linear or non-Gaussian. We propose an efficient particle Markov chain Monte Carlo algorithm to estimate the demographic parameters without the need for resorting to linear or Gaussian approximations. In particular, we exploit the integrated model structure to enhance the efficiency of the algorithm. We then incorporate the algorithm into a sequential Monte Carlo sampler in order to perform model comparison with regards to the dependence structure of the demographic parameters. Finally, we demonstrate the applicability and computational efficiency of our algorithms on two real datasets.

Both sequential Monte Carlo (SMC) methods (a.k.a. ‘particle filters’) and sequential Markov chain Monte Carlo (sequential MCMC) methods constitute classes of algorithms which can be used to approximate expectations with respect to (a sequence of) probability distributions and their normalising constants. While SMC methods sample particles conditionally independently at each time step, sequential MCMC methods sample particles according to a Markov chain Monte Carlo (MCMC) kernel. Introduced over twenty years ago in [6], sequential MCMC methods have attracted renewed interest recently as they empirically outperform SMC methods in some applications. We establish an $\mathbb{L}_r$ -inequality (which implies a strong law of large numbers) and a central limit theorem for sequential MCMC methods and provide conditions under which errors can be controlled uniformly in time. In the context of state-space models, we also provide conditions under which sequential MCMC methods can indeed outperform standard SMC methods in terms of asymptotic variance of the corresponding Monte Carlo estimators.

We present approximate algorithms for performing smoothing in a class of high-dimensional state-space models via sequential Monte Carlo methods ('particle filters'). In high dimensions, a prohibitively large number of Monte Carlo samples ('particles'), growing exponentially in the dimension of the state space, is usually required to obtain a useful smoother. Employing blocking approximations, we exploit the spatial ergodicity properties of the model to circumvent this curse of dimensionality. We thus obtain approximate smoothers that can be computed recursively in time and parallel in space. First, we show that the bias of our blocked smoother is bounded uniformly in the time horizon and in the model dimension. We then approximate the blocked smoother with particles and derive the asymptotic variance of idealised versions of our blocked particle smoother to show that variance is no longer adversely effected by the dimension of the model. Finally, we employ our method to successfully perform maximum-likelihood estimation via stochastic gradient-ascent and stochastic expectationmaximisation algorithms in a 100-dimensional state-space model.

The embedded hidden Markov models (EHMM) sampling method is a Markov chain Monte Carlo (MCMC) technique for state inference in non-linear non-Gaussian state-space models which was proposed in Neal (2003);Neal et al. (2004) and extended in Shestopaloff and Neal (2016). An extension to Bayesian parameter inference was presented in Shestopaloff and Neal (2013). An alternative class of MCMC schemes addressing similar inference problems is provided by particle Markov chain Monte Carlo (PMCMC) methods (Andrieu et al., 2009(Andrieu et al., , 2010. All these methods rely on the introduction of artificial extended target distributions for multiple state sequences which, by construction, are such that one randomly indexed sequence is distributed according to the posterior of interest. By adapting the Metropolis-Hastings algorithms developed in the framework of PMCMC methods to the EHMM framework, we obtain novel particle filter (PF)-type algorithms for state inference and novel MCMC schemes for parameter and state inference. In addition, we show that most of these algorithms can be viewed as particular cases of a general PF and PMCMC framework. We compare the empirical performance of the various algorithms on low-to highdimensional state-space models. We demonstrate that a properly tuned conditional PF with 'local' MCMC moves proposed in Shestopaloff and Neal (2016) can outperform the standard conditional PF significantly when applied to high-dimensional state-space models while the novel PF-type algorithm could prove to be an interesting alternative to standard PFs for likelihood estimation in some lower-dimensional scenarios.

The power–duration relationship describes the time to exhaustion for exercise at different intensities. It is generally believed to be a "fundamental bioenergetic property of living systems" that this relationship is hyperbolic. Indeed, the hyperbolic (a.k.a. critical-power) model which formalises this belief is viewed as the "gold standard" for assessing exercise capacity, e.g. in cycling, running, rowing, and swimming. However, the hyperbolic model is now the focus of two heated debates in the literature because: (a) it unrealistically represents efforts that are short (< 2 minutes) or long (> 15 minutes); (b) it contradicts widely-used performance predictors such as the so-called functional threshold power (FTP) in cycling. We contribute to both debates by demonstrating that the power–duration relationship is more adequately represented by an alternative, power-law model. In particular, we show that the often observed good fit of the hyperbolic model between 2 and 15 minutes should not be taken as proof that the power–duration relationship is hyperbolic. Rather, in this range, a hyperbolic function just happens to approximate a power law fairly well. We also prove a mathematical result which suggests that the power-law model is a safer tool for pace selection than the hyperbolic model. Finally, we use the power-law model to shed light on popular performance predictors in cycling, running and rowing such as FTP and Jack Daniels' "VDOT" calculator.