It helps to understand what changing the ISO setting in the camera does. The signal that is acquired by the sensor is amplified (typically on the sensor) by a variable gain amplifier and then sent to the A/D to be digitized, and the goal is to use the entire dynamic range of the A/D without exceeding it. Once you’ve done everything you can to capture as much light/photons (per ChgCanon’s list), your next step is to raise the ISO. Raising the ISO beyond base does one of two things - it either raises the gain of the variable gain amplifier (increasing the signal to the A/D) or it multiplies the digital output of the A/D by a constant. The changeover in method typically occurs around ISO 1000-1200, but depends on the specific camera. The advantage of using as much of the variable gain amplifier’s increase as possible is that you maximize the D/R of the A/D. For example, if the light is so low that at base ISO, the signal is only large enough to use the first 4 bits of a 12 bit A/D, then the max DR will only be 4 bits/stops (it will actually be less due to various digitizer errors). On the other hand, if you dial in enough amplification to just set the most significant bit of the A/D, you’ll approach the full 12 bits of DR available. Once you pass the point of max amplifier gain, further increases in ISO simply multiply the digital output of the A/D by a constant. That can be done in camera or in PP - the operation is the same, but I’d contend it makes more sense to do it in camera, simply because you rarely know exactly where the transition between the method occurs.

In my opinion, the concept of always shooting at base ISO and fixing in post on so called ISO invarient cameras is a mistake for the reasons stated above. In the example, if you only have 4 bits of resolution/DR, multiplying that small digital value doesn’t magically replace the missing data - it may give you the same maximum value (brightness), but all those missing real data points are now at best, just interpolated points. There are also other factors at play including noise generated by the sensor and amplifier as well as quantization and linearity errors of the A/D, but using all the D/R of the A/D is critical. And by all means, use all the steps in Chg Canon’s list to increase the actual exposure/light captured by the sensor, which is the first priority.