A few days ago I attended a symposium where the topics of obesity and probiotics were discussed.  It was an informative set of lectures.  Let me give you a few facts relating to these subjects.
Obesity in dogs is a primary precursor to torn ligaments in the knees of dogs.
The surgery to correct the torn ligament can be done a few ways, but arthritis will absolutely be a factor for the affected pet.
One of the primary modalities to address knee problems is weight loss.
Fat releases pro-inflammatory hormones.
The digestive tract is an important player in the body’s immune defense system.
Probiotics are live cultures of bacteria that promote proper digestive function.

Probiotics typically resolve diarrhea through regulation of the population of bacteria in the gut.