## 5 Discussion  
The genetic architecture of complex traits can well be understood by incorporating their underlying developmental features described by mathematical functions. Functional mapping that integrates genetics, statistics and developmental biology as a whole can be useful for deciphering the ontogenetic development of the genetic control of a complex quantitative trait [8; 9]. Original models for functional mapping were derived within the maximum likelihood (ML) context and implemented with the EM algorithm. A similar approach for mapping logistic QTLs was used by [PERSON] [8]. Although ML-based approaches possess many favorable statistical properties in parameter estimation, they may not be powerful enough to handle the complexity of high-dimensional QTL mapping models, as often seen in functional mapping. As an increasingly popular approach, Bayesian methods display remarkable capacity to estimate genetic parameters in QTL mapping [17; 18; 19; 39; 40]. Figure 4: Estimated marginal posterior distributions of the QTL location over a simulated linkage group with different matric-structuring approaches, unstructured (top), SAD(1)-structured (middle) and AR(1)-structured (bottom). In this article, we derived a general Bayesian framework for functional QTL mapping of dynamic traits and implemented Markov chain Monte Carlo (MCMC) algorithms to locate genomic positions of QTLs, and estimate the mathematical parameters that define a biological process and the statistical parameters that model the covariance structure. The Bayesian-based model allows the estimation of these parameters and their confidence intervals based on posterior distributions, and has great power to handle complex estimation issues related to functional mapping in an effective way. Like original parametric functional mapping [9], the new model allows to approximate the ontogenetic changes of the genetic effects triggered by a QTL. Because many biological processes, such as growth, follow a particular pattern of development, the ontogenetic control of a QTL can be mathematically described and, thereby, tested by estimating the parameters that define a biological process. The new model also take an advantage of functional mapping to model the structure of covariance matrix by a stationary or nonstationary approach. The application of Bayesian approaches to functional mapping was also considered by other authors. [PERSON] and [PERSON] [19] integrated Bayesian shrinkage approaches to map dynamic QTL, but their model was based on a nonparametric Legendre polynomial fitting. Although this treatment may be statistically flexible, its biological relevance may be limited because no biologically sensible functions are deployed. Also, [PERSON] and [PERSON]'s [19] approach did not utilize the high-efficiency characteristic of functional mapping through modeling the structure of an autocorrelated covariance matrix. Figure 5: The profile of the log-likelihood ratio (LR) test statistics between the full (there is a QTL) and reduced (there is no QTL) models across a simulated linkage group. The covariance matrix was structured by the SAD(1) model. Given a simulated time-dependent covariance matrix, Bayesian-based functional mapping obtains different results about the precision of parameter estimation, depending on whether and how such a covariance matrix is structured. This suggests the importance of estimating the residual covariance matrix effectively and efficiently. The unstructured approach captures the full information of time-dependent variances and covariances, which is effective but not efficient. The SAD(1)-structured model approximates the changes of variance and correlation over time [20], which is not only efficient (through estimating fewer parameters) but also effective (by reflecting the reality of this simulated data set to great extent). The AR(1)-structure model assuming the stationarity in both variance and correlation is efficient but not effective as much as the SAD(1) model.