From this link
The Hulse–Taylor system's orbit has decayed since the binary system was initially discovered, in precise agreement with the loss of energy due to gravitational waves. The ratio of observed to predicted rate of orbital decay is calculated to be 0.997±0.002.
The total power of the gravitational radiation (waves) emitted by this system presently, is calculated to be 7.35 × 1024 watts. For comparison, this is 1.9% of the power radiated in light by our own Sun.
So how do we know the Hulse–Taylor system's loss of energy is not (at least partially) due to electromagnetic radiation too? Especially if we don't even see the other pulsar due to its unfavorable inclination?