Compatibilists argue that determinism is compatible with human freedom, and that indeterminism is not compatible or at best incoherent. They feel (correctly) that there must be a deterministic or causal connection between our will and our actions. This allows us to take responsibility for our actions, including credit for the good and blame for the bad.
If the proximate cause of our actions was undetermined, the result of an uncaused quantum mechanical event in the mind, it would not be freedom of a kind worth having and we should disavow responsibility.
Most compatibilists accept the view of a causal chain of events going back indefinitely in time, consistent with the laws of nature, with the plan of an omniscient God, or with other determinisms. As long as our own will is included in that causal chain, we are free, they say. And they think causality in nature is related to the very possibility of reason and logic. Without causality, they say, we could not be certain of the truths of our arguments.
We think compatibilists should be classified according to the particular determinisms they think are compatible with human freedom. It is one thing to claim compatibility with modern physics, another to claim compatibility with God's foreknowledge.
Some compatibilists, often reluctantly, accept the view that random quantum mechanical events occur in the world. Whether in the physical world, in the biological world (where they are a key driver of genetic mutations), or in the mind, randomness and uncaused events are real. Quantum events introduce the possibility of accidents, novelty, and human creativity.
Our Cogito model assumes causality and adequate determinism in the critical apparatus of the Macro Mind. From the Micro Mind however, as well as from the external world including other minds, come surprising and unpredictable events to feed the Agenda of possible thoughts and actions. The Cogito is compatibile with both determinism and uncertainty. It lives in Eddington's "halfway house."
Compatibilism explains the will. It cannot explain free. Our new mind model gets us both free (random) and will (adequately determined). The Cogito is genuine free will.
For Teachers
Note there is also incompatibilism. There are two kinds of incompatibilists, those who deny human freedom (usually called "hard" determinists) and those who assert it (often called voluntarists, free willists, or metaphysical libertarians - to distinguish them from political libertarians).
For Scholars
The first compatibilist was Carneades (214-129), the great Skeptic.

Chapter 3.7 - The Ergod Chapter 4.2 - The History of Free Will
Part Three - Value Part Five - Problems