A Study of Baricitinib (LY3009104) in Participants From 2 Years to Less Than 18 Years Old With Active JIA-Associated Uveitis or Chronic Anterior Antinuclear Antibody-Positive Uveitis

Inclusion Criteria: * Participants must have a diagnosis of JIA-U or chronic anterior antinuclear antibody (ANA) positive uveitis without systemic features. * Participants must have active anterior uveitis, defined as cellular infiltrate in the anterior chamber of SUN criteria grade ≥1+ at screening and randomization, despite prior treatment with adequate doses of topical steroid therapy and methotrexate (MTX). * Participants must have an inadequate response or intolerance to MTX. * Participants must be on a stable dose of corticosteroid eye drops for at least 2 weeks prior to screening with a maximum of 4 drops/day/eye at screening. * Participants and their partners of child bearing potential must agree to use 2 effective methods of contraception for the duration of the study and for at least 1 week following the last dose of investigational product. Exclusion Criteria: * Participants must not have a history or presence of any autoimmune inflammatory condition other than JIA, such as Crohn's disease or ulcerative colitis. * Participants must not have any contraindications to adalimumab. * Exception: Participants who are biologic disease-modifying antirheumatic drug (bDMARD) inadequate response or intolerance with a contraindication to adalimumab may be enrolled, as they will be assigned to baricitinib. * Participants must not have increased intraocular pressure ≥25 millimeters of mercury (mm Hg) or that required treatment, including increases in medications, surgery, or hospitalization, within 4 weeks prior to baseline that, in the opinion of the investigator, would pose an unacceptable risk to the participant. * Participants must not have had intraocular surgery within the 3 months prior to screening (such as for cataract(s), glaucoma or vitrectomy). * Participants must not have a current or recent (\<4 weeks prior to baseline) infection. * Participants must not have a positive test for hepatitis B virus (HBV) at screening. * Participants must not have evidence of active tuberculosis (TB) or untreated latent TB.