If you spin a bucket (pail) of water, initially only the bucket spins, the water remains static and its surface flat. Soon, because of friction, the water spins too, and piles up around the edges, because inertia means it moves in a straight line, and outwards from the spin axis. Eventually, a concave water surface and a pressure gradient forms that balances the inertial tendency outwards. What does this inertial force depend upon? Not, noticeably, on motion relative to the bucket wall, because at first when the fluid was moving relative to the wall there was no curve in its surface. Newton concluded that there was only a curve when the water was moving relative to absolute space. Later, Berkeley and Mach said there was no such thing as absolute space (Einstein based relativity on this) and that the spin has to be measured relative to something observable like the fixed stars, so that a pail spinning in endless emptiness would not experience inertial forces at all. Einstein called this idea Mach's principle: inertia here is due to masses there. Considering the fixed stars also allowed Mach to wonder what would happen if Newton's bucket was static, but the fixed stars span around it. He said: "no one is competent to say how this experiment would turn out." (Bradley, 1971). Would the surface of the water curve or not?
MiHsC works in a similar way to Mach's Principle and states that the inertia of an object here increases with its acceleration relative to other masses there. It is predictive, so it has something new to say about Newton's bucket: if the fixed stars were whirled around it, there would be the same mutual acceleration between, each part of, the water and the stars, that occurs when the water spins and the stars are fixed. So, according to MiHsC, the inertial mass of the water would increase in the same way in both cases. By MiHsC & the conservation of momentum, in the bucket-spin case the initial spin would be slowed by the gain in inertia, and in the fixed-stars-spin case the water would start to rotate following the stars.
For observational evidence I can cite the flyby anomalies and the Tajmar experiment. With the flyby anomalies, small observed jumps in flyby spacecraft speed can be explained by MiHsC as being due to changes in the inertia of the craft as they accelerate relative to all the matter in the spinning Earth (McCulloch, 2008). In Martin Tajmar's experiments, a ring (instead of the fixed stars) was spun around a gyroscope (in a cold environment to purge all other accelerations), and the gyro followed the rotation of the ring (slightly), just as predicted by MiHsC: the gyro gains inertial mass when the ring accelerates, and to conserve momentum it has to spin with the ring (McCulloch, 2011). In Tajmar's experiment, there was also a parity violation which is predicted by MiHsC as being due to the Earth's spin relative to the fixed stars. The success of MiHsC in predicting these cases, shows that in low acceleration environments (cold or deep space) Mach's Principle may have been glimpsed.
Bradley, J., 1971. Mach's philosophy of science. The Athlone Press.
McCulloch, M.E., 2008. Modelling the flyby anomalies using a modification of inertia. MNRAS, 389(1), L57-60. http://arxiv.org/abs/0806.4159
McCulloch, M.E., 2011. The Tajmar effect from quantised inertia. EPL, 95, 39002. http://arxiv.org/abs/1106.3266