This circuit is operating at a low frequency (4 MHz) but it will show the interaction in the power supply.
First the circuit without decoupling.
The battery has RF in it as you can see.
The output doesn't look to bad does it? Well let's take a look at the decoupled circuit.
The decoupling did take the RF out of the battery.
It also lowered the output a little. Is that good or bad? If the coupling was positive feedback we could have been headed towards oscillations.
Here we see the battery current with the RF bypass. Better but can still be improved.
Now we add a bypass around the battery.
The battery current is pretty well a straight line with the decoupling. This will prevent the stages coupling through the power supply. Some will say the components are redundant and unnecessary but the proof is in circuit performance. Build the basic circuit and test it . Then add the bypasses and see if you can tell a difference.
The MMIC amp datasheets give some details of the filtering. I posted one a couple of post back.