Pitfalls of estimating the marginal likelihood using the modified harmonic mean

The modified harmonic mean is widely used for estimating the marginal likelihood. We investigate the empirical performance of two versions of this estimator: one based on the observed-data likelihood and the other on the complete-data likelihood. Through an empirical example using US and UK inflation, we show that the version based on the complete-data likelihood has a substantial bias and tends to select the wrong model, whereas the version based on the observed-data likelihood works well.