**Inflammation and Depression in People With HIV Study** This study aims to understand if inflammation affects the brain circuits related to pleasure and movement in people with HIV and depression. It will explore how an anti-inflammatory drug, *baricitinib*, might help. *HIV* is a virus that affects the immune system, and *depression* is a feeling of sadness that can affect daily life. *Anhedonia* means not being able to feel pleasure, and *psychomotor slowing* means moving or thinking slowly. In this 10-week study, 60 men and women with HIV and depression will receive either baricitinib or a placebo (a pill with no medicine) to see the effects on depression symptoms. Participants will undergo medical tests, including brain scans and optional spinal taps. **Key Points:** - Participation lasts about 3.5 months. - Participants will be randomly assigned to receive a real drug or a placebo. - Includes tests like *fMRI* (a brain scan) and optional spinal taps. Participants must be men or women with HIV and depression, aged 18 to 65, and not currently on certain medications. Women must have a negative pregnancy test. Participants will have access to medical assessments throughout the study.