  One of the scientific objectives of the Laser Interferometer Space Antenna
(LISA) is to probe the expansion of the Universe using gravitational wave
observations. Indeed, as gravitational waves from the coalescence of a massive
black hole binary (MBHB) carry direct information of its luminosity distance,
an accompanying electromagnetic (EM) counterpart can be used to determine its
redshift. This method of $bright$ $sirens$, when applied to LISA, enables one
to build a gravitational Hubble diagram to high redshift. In this work, we
forecast the ability of LISA-detected MBHB bright sirens to constrain
cosmological models. The expected EM emission from MBHBs can be detected up to
redshift $z\sim 7$ with future astronomical facilities, and the distribution of
MBHBs with detectable counterpart peaks at $z\sim 2-3$. Therefore, we propose
several methods to leverage the ability of LISA to constrain the expansion of
the Universe at $z\sim 2-3$, a poorly charted epoch in cosmography. We find
that the most promising method consists in using a model-independent approach
based on a spline interpolation of the luminosity distance-redshift relation:
in this case, LISA can constrain the Hubble parameter at $z\sim2-3$ with a
relative precision of at least $10\%$.
