My answer would be: it depends. What is the nutrient content of the compost? What crop are you planning to grow? What is the past history of the bed: crops grown, amendments applied, soil type, cover crops if any? The best way to find the nutrient content of the compost is to have it tested by a soil lab, but in the meantime here's a link to a study from Penn State giving some typical values: http://www.mushroomcompost.org/files/theme/NPK2(1).pdf Which crop you're planning to grow has a big influence on how much nutrients are needed. Here's a link to an NRCS publication on nutrient management that gives requirements for several common vegetable crops (and will also give help with the math to figure out how much compost to use): http://www.Nutrient-Management-in-Organic-Systems-Western-States-Implementation-Guide%20(3).pdf This NRCS book will also show how to figure out any nutrient credits you have based on the past history of the bed. For example, your mushroom compost will continue releasing available nitrogen at a gradually decreasing rate for several years afte r you apply it. The nutrient values in a lab report or the Penn State report are percent by weight. To figure out what depth of compost to apply (which is a volume) you need to relate weight to volume. This is what bulk density does (weight/volume). In the Penn State report they have done this for you, giving N, P, and K in both % by weight and pounds/cubic yard.
browntown, I'm sure it depends on the analysis, as stated above, as well as your soil type, condition, etc...but I believe the general rule of thumb is that initially 3 to 4 inches. From there on out, you will just be applying as needed based on the crop (i.e. heavy, light feeder, cover) as a top dressing along with other amendments.