  The gravitational waveform of a merging stellar-mass binary is described at
leading order by a quadrupolar mode. However, the complete waveform includes
higher-order modes, which encode valuable information not accessible from the
leading-order mode alone. Despite this, the majority of astrophysical
inferences so far obtained with observations of gravitational waves employ only
the leading order mode because calculations with higher-order modes are often
computationally challenging. We show how to efficiently incorporate
higher-order modes into astrophysical inference calculations with a two step
procedure. First, we carry out Bayesian parameter estimation using a
computationally cheap leading-order-mode waveform, which provides an initial
estimate of binary parameters. Second, we weight the initial estimate using
higher-order mode waveforms in order to fold in the extra information from the
full waveform. We use mock data to demonstrate the effectiveness of this
method. We apply the method to each binary black hole event in the first
gravitational-wave transient catalog GWTC-1 to obtain posterior distributions
and Bayesian evidence with higher-order modes. Performing Bayesian model
selection on the events in GWTC-1, we find only a weak preference for waveforms
with higher order modes. We discuss how this method can be generalized to a
variety of other applications.
