All physical things follow certain rules of cause and effect. One event causes another event, which causes another. These sequences of cause and effect are called causal chains. Science can explore the causal chains by altering the cause and measuring what happens to the effect. If we can find things that don't follow those laws, then those things are not part of the physical world.
The subjective experience of color, sounds, pain, and so forth are caused by physical events. But they are outside the realm of what can be measured. (For example, there is no way, even in principle, to know whether my sensation of red is the same as your sensation of red.) This is because the experiences don't have a causal effect on the world which could be measured.
On the other side, free will choices can not be caused by anything, but they do have effects on the world. If subjective experiences are the ends of causal chains, free will choices are the beginnings.
It wouldn't be hard to build a machine that replaced subjective experiences and free will with a causal chain connecting the incoming and outcoming chain together. If the chain was complex enough or contained random elements, an outside observer couldn't know whether the system was causally connected or not. But we know from our own experience that that isn't the way it works within our own minds. The internal subjective experience would be absent in such a system.
Some people would claim that we can't be sure that such a system doesn't have a subjective experience. But I think we can be sure. There are parts of our own brains' workings that are inaccesible to consciousness. If we build a system that works in the same way as these parts, we can be certain that it won't have conscious internal experiences or free will.
Subjective experiences and the free will choices are connected. It is the presence of a conscious field of experience that makes free will choices possible. Our choices could simply be choices of which experiences to give more weight in the determination of an effect, by freely paying attention to certain aspects of the conscious field more than others. Every choice we make is a creative artwork, where we put the emphasis on what we decide is important.
It's analogous to a video game. Within the computer, everything happening in the game world is deterministic. But the causal chain ends in a display of colors on the monitor. New causal chains begin within the world of the game with the movement of the controller. But we sit outside the world. The colors on the screen don't cause the movements of the controller, but they provide the environment in which coherent choices of how to move the controller can occur.
There is another part to this connection between experiences and will. When we make a difficult choice ("exercising" the will), the choice is difficult because of the pain or discomfort it causes. When people deserve praise for having made good choices, it is because they did the right thing in spite of the pain it caused them. Without the subjective sensation of pain, there could be no working of the will.
The key to all this is that the choices are neither random nor caused. They are influenced by all kinds of things, but in the end are capable of working opposite to every influence.
The discoveries of the last century in quantum mechanics, chaos theory, and algorithmic information theory make such a world seem more possible and reasonable than in the period since Newton where science only knew of the more familiar kinds of mechanical causes. Quantum mechanics tells us that there will always be uncaused events in any system. Scientists believe uncaused quantum events to be completely random, but it is a gap in the system where subtle causes could sneak in. Chaos theory tells us that such minute causes could be amplified into macroscopic effects on the entire brain and from there, the outside world. In fact, neural networks are ideal for such amplification effects. From AIT, we know that even in mathematics, there are effects without causes.
It is reasonable to believe in such a system outside the physical world. We have internal evidence for it that is compelling. It requires the convoluted and unbelievable warping of the evidence to sustain a picture of the world that doesn't include such a supernatural world.
Notes for those with a background in philosophy: by "free will" I mean libertarian free will. By "subjective experience" I mean qualia. In this post I am assuming both as given from experience.