The marginal likelihood is commonly used for comparing different evolutionary models in Bayesian phylogenetics and is the central quantity used in computing Bayes Factors for comparing model fit. A popular method for estimating marginal likelihoods, the harmonic mean (HM) method, can be easily computed from the output of a Markov chain Monte Carlo analysis but often greatly overestimates the marginal likelihood. The thermodynamic integration (TI) method is much more accurate than the HM method but requires more computation. In this paper, we introduce a new method, steppingstone sampling (SS), which uses importance sampling to estimate each ratio in a series (the "stepping stones") bridging the posterior and prior distributions. We compare the performance of the SS approach to the TI and HM methods in simulation and using real data. We conclude that the greatly increased accuracy of the SS and TI methods argues for their use instead of the HM method, despite the extra computation needed.

Osteocytes represent the most abundant cellular component of mammalian bones with important functions in bone mass maintenance and remodeling. To elucidate the differential gene expression between osteoblasts and osteocytes we completed a comprehensive analysis of their gene profiles. Selective identification of these two mature populations was achieved by utilization of visual markers of bone lineage cells. We have utilized dual GFP reporter mice in which osteocytes are expressing GFP (topaz) directed by the DMP1 promoter, while osteoblasts are identified by expression of GFP (cyan) driven by 2.3kb of the Col1a1 promoter. Histological analysis of 7-day-old neonatal calvaria confirmed the expression pattern of DMP1GFP in osteocytes and Col2.3 in osteoblasts and osteocytes. To isolate distinct populations of cells we utilized fluorescent activated cell sorting (FACS). Cells suspensions were subjected to RNA extraction, in vitro transcription and labeling of cDNA and gene expression was analyzed using the Illumina WG-6v1 BeadChip.Following normalization of raw data from four biological replicates, 3444 genes were called present in all three sorted cell populations: GFP negative, Col2.3cyan + (osteoblasts), and DMP1topaz + (preosteocytes and osteocytes). We present the genes that showed in excess of a 2-fold change for gene expression between DMP1topaz + and Col2.3cyan + cells. The selected genes were classified and grouped according to their associated gene ontology terms. Genes clustered to osteogenesis and skeletal development such as Bmp4, Bmp8a, Dmp1, Enpp1, Phex and Ank were highly expressed in DMP1topaz + cells. Most of the genes encoding extracellular matrix components and secreted proteins had lower expression in DMP1topaz + cells, while most of the genes encoding plasma membrane proteins were increased. Interestingly a large number of genes associated with muscle development and function and with neuronal phenotype were increased in DMP1topaz + cells, indicating some new aspects of osteocyte biology. Although a large number of genes differentially expressed in DMP1topaz + and Col2.3cyan + cells in our study have already been assigned to bone Contact Information: Ivo Kalajzic, Department of Reconstructive Sciences, MC 3705, University of Connecticut Health Center, 263 Farmington Ave., Farmington, CT 06032. Tel.: 860-679-6051; Fax: 860-679-2910; ikalaj@neuron.uchc.edu. Publisher's Disclaimer: This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. NIH Public Access Author ManuscriptBone. Author manuscript; available in PMC 2010 October 1. NIH-PA Author ManuscriptNIH-PA Author Manuscript NI...

Bayesian phylogenetic analyses often depend on Bayes factors (BFs) to determine the optimal way to partition the data. The marginal likelihoods used to compute BFs, in turn, are most commonly estimated using the harmonic mean (HM) method, which has been shown to be inaccurate. We describe a new more accurate method for estimating the marginal likelihood of a model and compare it with the HM method on both simulated and empirical data. The new method generalizes our previously described stepping-stone (SS) approach by making use of a reference distribution parameterized using samples from the posterior distribution. This avoids one challenging aspect of the original SS method, namely the need to sample from distributions that are close (in the Kullback–Leibler sense) to the prior. We specifically address the choice of partition models and find that using the HM method can lead to a strong preference for an overpartitioned model. In contrast to the HM method and the original SS method, we show using simulated data that the generalized SS method is strikingly more precise (repeatable BF values of the same data and partition model) and yields BF values that are much more reasonable than those produced by the HM method. Comparisons of HM and generalized SS methods on an empirical data set demonstrate that the generalized SS method tends to choose simpler partition schemes that are more in line with expectation based on inferred patterns of molecular evolution. The generalized SS method shares with thermodynamic integration the need to sample from a series of distributions in addition to the posterior. Such dedicated path-based Markov chain Monte Carlo analyses appear to be a cost of estimating marginal likelihoods accurately.

The inherent heterogeneity of bone cells complicates the interpretation of microarray studies designed to identify genes highly associated with osteoblast differentiation. To overcome this problem, we have utilized Col1a1 promoter-green fluorescent protein transgenic mouse lines to isolate bone cells at distinct stages of osteoprogenitor maturation. Comparison of gene expression patterns from unsorted or isolated sorted bone cell populations at days 7 and 17 of calvarial cultures revealed an increased specificity regarding which genes are selectively expressed in a subset of bone cell types during differentiation. Furthermore, distinctly different patterns of gene expression associated with major signaling pathways (Igf1, Bmp, and Wnt) were observed at different levels of maturation. Some of our data differ from current models of osteoprogenitor cell differentiation and emphasize components of the pathways that were not revealed in studies based on a total cell population. Thus, applying methods to generate more homogeneous populations of cells at a defined level of cellular differentiation from a primary osteogenic culture is feasible and leads to a novel interpretation of the gene expression associated with increasing levels of osteoprogenitor maturation.

Longitudinal studies of ageing make repeated observations of multiple measurements on each subject. Change point models are often used to model longitudinal data. We demonstrate the use of Bayesian and profile likelihood methods to simultaneously estimate different change points in the longitudinal course of two different measurements of cognitive function in subjects in the Bronx Aging Study who developed Alzheimer's disease (AD). Analyses show that accelerated memory decline, as measured by Buschke Selective Reminding, begins between seven and eight years before diagnosis of AD, while decline in performance on speeded tasks as measured by WAIS Performance IQ begins slightly more than two years before diagnosis, significantly after the decline in memory.