Yoga versus Pilates, which is safer?

For any exercise safety depends upon how the exercise is being taught as well as the technique performed by the participant.

I believe that inherently yoga has more risk for injury due to the following reason, when performing yoga you are asked to move your body without stabilization. Because of this your body will feed into it's compensations, the areas of the body that move too much.  To attain the range of motion needed for yoga these movements will increase instability in an already unstable area.

With Pilates you are instructed to brace various muscles and lengthen as you move. This provides protection to unstable joints. 

Lengthening (traction) as well as approximation (shortening) cause irradiation into the core muscles as does rotational movement. Rotational movement is incorporated into many of Pilates exercises. Additionally, Pilates equipment provides both approximation and traction.

I believe that the machines used in Pilates are more protective and beneficial than the Pilates mat work, especially for a novice. 

My feeling as a clinician is that people should choose to do the exercise that they enjoy. 

Here are several modifications that can be incorporated into yoga to make it safer and more beneficial.

1. When performing standing poses, push your feet into the floor and rotate them either out or in isometrically (without movement of the legs). Choose the direction that feels most comfortable
2. When performing poses where your hands are on the floor, keep shoulder blades down and rotate your hands either outward or inward isometrically (without movement of the arms). Choose the direction that feels most comfortable.
3. When you lift your arms keep your shoulder blades down as if to place your shoulder blades into your back pockets. 
4. Lengthen your body and limbs when moving through your poses. 
5. Maintain normal breathing. 
6. Move through your poses slowly and deliberately.