Therefore, the marginal likelihood P (F | Θ ) = ∑ H, Ψ P (F, H, Ψ | Θ ) is symmetric for the two parameter sets: P (F | Θ ¯ ) = P (F | Θ ).
Under M2, the marginal likelihood p (x ∣ M2) does not have a closed form, so we use a two-dimensional Monte Carlo integration to evaluate the marginal likelihood.
To extend this to multiple lineages, we use the marginal likelihood p(Di | Hj) to define the posterior probabilities of the different hypotheses Hj.
The area under the curve (AUC) values for the best performing DAGs were calculated using posterior probability P(G| D) with informative priors (proposed method) and marginal likelihood P(D| G) scores with uniform flat priors.
The problem can be seen as a model selection problem, where different comparisons are thought of as different model structures (H1,… H5) and, given experimental lineage commitment profile data D, the marginal likelihood P(D | Hj), j=1,…,5, is used to score different models.
Under M1, the marginal likelihood p (x ∣ M1) has a closed form: n 1 x 1 n 2 x 2 B (x 1 + x 2 + 1, n 1 + n 2 - x 1 - x 2 + 1 ), where B is the beta function.
If we consider a set of alternative models M i, i=1,…, N and do not assume any a priori preference for any of the models, the marginal likelihoods p(D| M i ) can be directly used to assess the evidence for different models.
By using the likelihood in Equation (9), we can write the marginal likelihood (10) p (y | X, θ, k ) = ∫ p (y | f, X, θ, k ) p (f | X, θ, k ) d f, where we have marginalized over the possible realizations of F. In this case, the integral in Equation (10) is not analytically tractable and we resort to Markov chain Monte Carlo (MCMC) methods.
In the Bayesian framework, model selection is closely related to parameter estimation, but the focus shifts onto the marginal posterior probability of model m given data D0, where P D0| m) is the marginal likelihood and P(m) the prior probability of the model (Gelman et al., 2003).
In this article, a ≡ Δ, b ≡ β, with independent priors p a, b) = p(a) p(b); additionally, due to INLA, deterministic approximations are available for both the conditional effects posterior p(b| y, a) and the conditional marginal likelihood p y| a).
When using the uniform model prior, a Bayes factor can be computed as the ratio of marginal data likelihoods: p(D | M i ) /p(D | M0), in which M i denotes any model with at least one change point while M0 represents the model under the null hypothesis, i.e., a model fitting a constant line to the gene expression profile.
