We discuss model selection to determine whether the variance-covariance matrix of a multivariate Gaussian model with known mean should be considered to be a constant diagonal, a non-constant diagonal, or an arbitrary positive definite matrix. Of particular interest is the relationship between Bayesian evidence and the flexibility penalty due to Priebe and Rougier. For the case of an exponential family in canonical form equipped with a conjugate prior for the canonical parameter, flexibility may be exactly decomposed into the usual BIC likelihood penalty and a $O_p(1)$ term, the latter of which we explicitly compute. We also investigate the asymptotics of Bayes factors for linearly nested canonical exponential families equipped with conjugate priors; in particular, we find the exact rates at which Bayes factors correctly diverge in favor of the correct model: linearly and logarithmically in the number of observations when the full and nested models are true, respectively. Such theoretical considerations for the general case permit us to fully express the asymptotic behavior of flexibility and Bayes factors for the variance-covariance structure selection problem when we assume that the prior for the model precision is a member of the gamma/Wishart family of distributions or is uninformative. Simulations demonstrate evidence's immediate and superior performance in model selection compared to approximate criteria such as the BIC. We extend the framework to the multivariate Gaussian linear model with three data-driven examples.

翻译：我们讨论模式选择,以确定具有已知平均值的多变量高斯模型差异变量矩阵是否应被视为一个常态对角、非常态对角对角或任意正确定矩阵。 特别感兴趣的是巴伊西亚证据和因普里贝和鲁吉埃而导致的灵活性罚款之间的关系。 对于在卡文参数之前配有共产体的卡文形式指数式家庭来说,灵活性可能完全分解为通常的BIC概率罚款和美元O_p(1)美元,而后者是我们明确计算的。 我们还调查了配有相近的线型卡尼西亚指数型家庭对巴伊斯系数因素的反射率; 特别是,我们发现贝斯因素的精确率有利于正确的模型: 在完整和嵌入模型时, 灵活性可能完全分解为BIC概率和美元($O_p(1)美元)的术语。 普通案例的理论考虑使我们能够充分表达“不稳度”模型的不稳度行为模式和“巴伊因子”的精确度指数性, 也就是我们所选择的精确度结构中, 精确度的精确度是我们所选取的精确度标准, 。