What is the effectiveness of self-prescribed marijuana for depression? How common is it for patients diagnosed with depression to self-prescribe themselves marijuana after trying prescription anti-depressants? Is there any evidence that this self-treatment is effective or satisfies some need that is not filled with typical prescription medication?
Question 1: I am unaware of any studies directly addressing the epidemiology of self-medication with cannabis in cases of treatment-resistant depression (it seems like that is what you are asking about). However, there are several lines of therapy that have demonstrated effectiveness in depression beyond the first line antidepressants (CANMAT Depression Guidelines). None of these include marijuana or cannabinoids.

Question 2: There is (to my knowledge) absolutely no evidence that marijuana is an effective treatment for depression. The article by Volkow et al. (2014) although titled "Adverse Effects," also reviews conditions for which evidence of effectiveness exists for symptomatic treatment with marijuana or cannabinoids. These conditions include glaucoma, nausea, AIDS-associated anorexia and wasting syndrome, chronic pain, inflammation, multiple sclerosis, and epilepsy (Volkow et al. 2014). That being said, the quality of that evidence is unclear to me. 

It must also be mentioned that studies will generally be limited by heterogeneous cannabinoid content in the active compound. Not all cannabinoids are created equal, and from what I understand one of the potentially most medically useful cannabinoids is cannabidiol (CBD; Curran et al., 2016); THC and CBD may have opposing effects, with THC generally associated with impairments across several psychological functions (Curran et al. 2016). On the other hand, small studies have suggested a potential role for CBD (but not THC) in augmenting psychotherapeutic treatment of some anxiety disorders. The evidence thus far seems to suggest that marijuana would not have a role in treatment of psychiatric disorders in general, let alone depression, but that further research about the effects of isolated CBD might be warranted.

Reference


Volkow, N. D., et al. (2014). Adverse Health Effects of Marijuana Use. New England Journal of Medicine, 370(23), 2219–2227.    
Curran, H. V, Freeman, T, P., Mokrysz, C., Lewis D, A., Morgan, C, J, A., &amp; Parsons, L. H. (2016). Keep off the grass? Cannabis, cognition and addiction. Nature Reviews Neuroscience, 17(5), 293–306.   
Baldwin, D. S., Anderson, I. M., Nutt, D. J., Allgulander, C., Bandelow, B., den Boer, J. a, … Wittchen, H.-U. (2014). Evidence-based pharmacological treatment of anxiety disorders, post-traumatic stress disorder and obsessive-compulsive disorder: a revision of the 2005 guidelines from the British Association for Psychopharmacology. Journal of Psychopharmacology (Oxford, England), 28, 403–39.