The burden of proof is a key feature of legal trials. In brief, in order to secure a conviction in a criminal case or an award of money damages in a civil case, the moving party must produce sufficient evidence or proof that his allegations are true. (In the Anglo-American legal tradition, the burden is “proof beyond a reasonable doubt” in criminal cases, while the “preponderance of the evidence” standard is used in most civil cases.) This concept is relevant to many areas of life beyond law as well. Here is Google Scholar; below is a partial lit review:
J. P. McBaine, Burden of proof: degrees of belief, California Law Review, Vol. 32, No. 3 (1944), pp. 242-268, available here. This law review article contains proposed jury instructions for various burdens of proof (i.e. preponderance of the evidence, clear & convincing proof, and proof beyond a reasonable doubt.
Rita James Simon & Linda Mahan, Quantifying burdens of proof: a view from the bench, the jury, and the classroom, Law & Society Review, Vol. 5, No. 3 (1971), pp. 319-330, available here. This paper presents the results of several experiments conducted on sociology students at the University of Illinois and on judges and jurors in Champaign County Court. Specifically, the researchers attempt to measure how students, judges, and jurors translate legal burdens of proof into numerical probability estimates.
Louis Kaplow, Burden of proof, Yale Law Journal, Vol. 121, No. 4 (2012), pp. 738-859, available here. This book-length “paper” (consisting of 122 pages and 218 footnotes) explores the burden of proof from an economic perspective.