For each treatment (column 1), the two runs of highest harmonic-mean marginal likelihood are reported and compared using Bayes Factors as calculated using 1000 bootstrap pseudoreplicates in Tracer v.1.5, which employs a weighted likelihood bootstrap estimator.
"What do you mean marginal benefits?" he asked.
Here, X is matrix of gene expression data, P(X | G) means the (marginal) likelihood probability, and P G) means a prior distribution over network structure G. Based on the above discussion, the combined score measure consists of two parts: one is the approximate likelihood, and the other is the network prior information.
The best model is the one that gives the lowest BIC value, which means the largest marginal likelihood of data.
When modeling QTL affecting the mean only, the marginal likelihood for observation i (omitting covariates) is the mixture L i = ∑ j = 1 3 p i j N (r j α, σ 2 ), where p i = (p i1, p i2, p i3) is defined as above for two founder lines, N is the normal density, and r = (–1, 0, 1).
At the same time dividing the dataset into 3 partitions seemed more logical if harmonic means of marginal likelihoods were compared (-455638.67, -455391.40, -449073.19 for 61, 4 and 3 partitions, respectively).
After a run of 1 million iterations in BayesTraits [15] to estimate the correlation between leaf-living ecology and reduced morphology, the following harmonic means of the marginal likelihood were obtained, with virtually no difference between runs: ∼71.50 for the model in which leaf-living ecology and reduced morphology are independent, and ∼56.50 for the model in which they are dependent.
Furthermore, the use of the marginal likelihood means our method inherits the ease with which side information can be incorporated in probabilistic models.
From now on, when we refer to the marginal likelihood, we mean the marginal likelihood for a specific model and we forego conditioning on the model Mk in the notation.
The harmonic mean estimator of the marginal likelihood is only decisive when the ln(BF) is > 4.6.
