Combining principal component analysis with parameter line-searches to improve the efficacy of Metropolis–Hastings MCMC

Abstract

When Markov chain Monte Carlo (MCMC) algorithms are used with complex mechanistic models, convergence times are often severely compromised by poor mixing rates and a lack of computational power. Methods such as adaptive algorithms have been developed to improve mixing, but these algorithms are typically highly sophisticated, both mathematically and computationally. Here we present a nonadaptive MCMC algorithm, which we term line-search MCMC, that can be used for efficient tuning of proposal distributions in a highly parallel computing environment, but that nevertheless requires minimal skill in parallel computing to implement. We apply this algorithm to make inferences about dynamical models of the growth of a pathogen (baculovirus) population inside a host (gypsy moth, Lymantria dispar). The line-search MCMC appeal rests on its ease of implementation, and its potential for efficiency improvements over classical MCMC in a highly parallel setting, which makes it especially useful for ecological models.

Appendices

Appendix 1: Sampling–importance–resampling

Directly simulating many realizations of a birth–death process is computationally expensive. To avoid this cost for the linear birth–death model, we instead sample directly from the distribution of first passage times, using a sampling-importance-resampling algorithm. This method is possible because the function that describes the first passage time for a linear birth–death model can be evaluated point-wise (Shortley 1965).

We began our algorithm, for a given parameter set, by first numerically integrating the first-passage-time function in the C programming language, using the ‘gsl_integration_qag’ function from the GNU Scientific Library, over the range $[0, 612]$, matching the range of observation times in our experiment. Because the linear birth–death model has an absorbing boundary if the population size hits zero, not every trajectory will cross the upper threshold that leads to host death, and so the integral of this function will be in the range $[0,1]$, with the integral value, $p_d$, corresponding to the probability of host death occurring by hour 612. For a given number of model trajectories $\nu$, the number of host deaths is then a number drawn from a binomial distribution with parameters $\nu$ and $p_d$.

To generate these first passage times from our target distribution we used a sampling-importance-resampling algorithm. First, we generated $10^4$ potential first passage times, $u_i$, from a uniform distribution on the interval $[0, 612]$. This interval was chosen so that these points would span the range of our data. Second, we calculated weights $W(u_i)$ for each of these time points, using the density function for first passage time $Q_d(\cdot )$ proposed by Shortley (1965). Weights were thus calculated as

$$\begin{aligned} W(u_i) = Q_d(u_i)/ \underset{u_i}{\text {argmax}}(Q_d(u_i)). \end{aligned}$$

(17)

Third, we generated the first passage times from our target distribution by resampling $u_i$ according to the respective weights $W(u_i)$.

Appendix 2: Sensitivity to priors

In the main text of the paper, we ran our MCMC routine using improper priors, which can sometimes lead to improper posterior distributions. We believe that an improper posterior is unlikely in our case, because each of our multiple MCMC chains seem to have converged on the same stationary distribution. As an additional test, however, we further examined the model behavior under different sets of priors. To do this, we re-ran our analysis using half-normal priors that are vague but proper. Thus for each parameter

$$\begin{aligned} \pi (\theta ) = \left\{ \begin{array}{ll} \frac{2}{\sqrt{2\pi \times 10^{14}}} e^{-\frac{\theta ^2}{2 \times 10^{14}}} &{}\quad \text{ if } \ \theta \ge 0;\\ 0 &{}\quad \text{ otherwise }.\end{array} \right. \end{aligned}$$

(18)

where $\theta$ is defined as $\beta , \phi , c_1, c_2, \text {or } m$.

We also re-ran this analysis using a set of half-normal priors similar to the above, but with parameter-specific variances for the priors of $\beta$, $\phi$, $c_1$, $c_2$, and $m$ of $10^0$, $10^0$, $10^4$, $10^6$, and $10^{14}$ respectively.

From this analysis, we observe that the resulting marginal posteriors are very similar to those achieved in our earlier analysis using improper priors (Fig. 10), providing additional evidence that the data are informative about the model parameters, and that the results are robust to the choice of priors.

Fig. 10

Sensitivity of posterior to priors. Posterior distributions for $log_{10}$ values of each parameter are shown above. The black line shows the posterior distribution using the improper priors from the main text. The red line shows the posterior distribution when using the vague proper priors in Eq. (18). The blue line shows the posterior distribution when using the parameter-specific variances given in the text. We achieve very similar posterior distributions using each sets of priors (Color figure online)

Appendix 3: Bias in posterior estimates

Although the MCMC routine used in this paper appears to converge to a stationary distribution, the distribution is not exactly equal to the posterior distribution. Proposed parameter sets can be accepted at an inflated rate, because of uncertainty in our estimate of the likelihood, and our MCMC chains tend to over-accept proposed jumps. The result of this over-acceptance is a stationary distribution that is biased towards the proposal distribution.

The uncertainty in our estimate of the likelihood depends on the number of realizations used to parameterize Eq. (6), and so an obvious way to eliminate this bias would be to increase the number of realizations. Our precision is thus directly related to computing time, and so in the face of limited computing resources, we are forced to allow for at least some bias. At the number of realizations we used ($3\times 10^3$), however, the bias in our realized posterior distribution is minimal. To show this, we re-ran our analyses for a range of numbers of realizations. As Fig. 11 demonstrates, increasing the number of realizations at first leads to dramatic changes in the posterior, but as we approach $3\times 10^3$ realizations, further increases have essentially no effect. This suggests that our stationary distribution is probably close to the true posterior distribution.

Fig. 11

Bias in the realized posterior distribution. Posterior distributions for $log_{10}$ values of each parameter are shown above. Each line shows the results when using a particular number of realizations to estimate the likelihood, with the solid black line showing the posterior distribution at the number of realizations used in the paper. By comparing the change in the posterior distribution at different numbers of realizations, we get a sense of how bias is affecting our results. While bias seems to be important at low numbers of realizations, at higher numbers the distributions seem to stabilize, suggesting that little bias remains at $3\times 10^3$ realizations

Appendix 4: Implications of the results for the nonlinear dynamical model

Our estimates of the parameters of the nonlinear model are listed in Table 2, but here we place these estimates in the context of baculovirus biology. First, our doubling-time estimate of 3.04 h is similar to a doubling-time estimate of 2.53 h for the cabbage looper Trichoplusia ni calculated using DNA-DNA hybridization (Beek et al. 1990). We did not necessarily expect close agreement between these estimates because the two insects and their associated baculoviruses are not closely related, but the rough similarity suggests that our estimate is biologically reasonable.

Table 2 Parameter estimates

Second, our estimate of the half-saturation constant of the virus-dose function is $c_{2} \approx 10^{3}$, which is much lower than the $2 \times 10^{9}$ virus particles that are produced by a virus-killed, fourth-instar gypsy-moth cadaver (Shapiro et al. 1986). It thus appears that virus doses in nature are nearly saturated, so that small changes in dose have little effect on host times of death. This is surprising because virus strains could presumably kill faster if they produced fewer virus particles. We would therefore expect that natural selection would favor virus strains with shorter speeds of kill, because the cost of producing fewer virus particles appears to be very low. In nature, however, the virus is rapidly rendered inactive by ultraviolet light (Fuller et al. 2012), and so consumed doses of infectious virus may often be quite small. The slow speed of kill of this virus may therefore be an adaptation to high virus-inactivation rates, because slow-killing virus strains produce large numbers of particles that help to reduce the risk that all particles will be inactivated (Shapiro et al. 2002). Our estimate of $c_{2}$ therefore suggests that selective forces acting within hosts may oppose selective forces acting between hosts, as has often been suggested by mathematical theories of pathogen evolution (Antia et al. 1994; Gilchrist and Sasaki 2002).

Our best estimate of the largest average number of virus particles that could initiate an infection is $c_{1} \approx 35$. Given that the highest virus dose used was $1.35 \times 10^4$ particles, our estimate of $c_{1}$ suggests that the vast majority of consumed virus particles play no role in infection, even though larvae almost certainly have many more than $35$ midgut epithelial cells (Baldwin and Hakim 1991). This observation can be explained by cell sloughing, in which cells of the larval midgut are removed and subsequently replaced by new cells (Baldwin and Hakim 1991). Our estimate of $c_{1}$ thus supports previous research suggesting that cell sloughing is an important line of defense against baculovirus infection (McNeil et al. 2010; Hoover et al. 2000). Our estimate of $c_1$ also implies that severe population bottlenecks occur at the beginning of each new infection, in turn suggesting that genetic drift may be an important evolutionary force sha** the virus population.

Our estimate of the number of immune cells in a healthy larva is $m = 7 \times 10^4$. Examination of the posterior distribution revealed that this estimate is actually highly uncertain, because of a strong, negative log-linear correlation with the immune-cell attack rate $\beta$ (Fig. 8). The strong correlation between these two parameters might be expected given that in the deterministic version of the model, these two parameters are individually non-identifiable.

Mechanistic models of within-host pathogen growth have a long history (Antia et al. 1994; Alizon and Baalen 2008; Shortley 1965), but few of these models have been challenged with data, because of the computational difficulties associated with fitting nonlinear, dynamic models. Although fitting static or deterministic models to response data has provided useful insights into the infection process of some pathogens (Meynell 1957), including baculoviruses (Beek et al. 2000; Zwart et al. 2009), a growing literature strongly suggests that within-host pathogen population growth is stochastic (Grant et al. 2008; Kennedy et al. 2014; Vaughan et al. 2012). Incorporating this stochasticity and using the entire distribution of speeds of kill to make inference is superior to basing the inference simply on the mean quantities. Our work therefore demonstrates the usefulness of nonlinear stochastic models in understanding within-host pathogen growth. Moreover, nonlinear dynamic models are becoming increasingly popular in ecology, highlighting the need for easy-to-implement statistical algorithms suitable for use with such models.

For baculoviruses in particular, survival-time data are widely available, but are usually used only to calibrate parametric phenomenological models such as those based on the Weibull distribution (Mudholkar et al. 1996; Morgan 1992). By instead using speed-of-kill data to fit a more mechanistic model, we have gained useful insights into the underlying biological processes, which in turn has allowed us to make inferences about virus evolution. In particular, our results suggest that genetic drift likely plays an important role in the evolution of the virus, which is important partly because drift may oppose the effects of natural selection (Kimura 1983). The occurrence of drift also has implications for the use of baculoviruses in pest control, because control programs often use only a single strain of virus (Hunter-Fujita et al. 1998). This has led to concerns that virus sprays will reduce natural diversity, and our results suggest that such reductions may be exacerbated by the drift inherent in the infection process.