This paper reviews recent contributions from a Bayesian-oriented perspective, after the ASA statement on p-values (2016). We classify proposals that (i) supplement the p-value; (ii) modify the p-value itself. In the first group, we review the Bayes factor, the False Positive risk, the rejection odds and the analysis of credibility from both Matthews\' and Held\'s point of view. We also put forth and discuss a new index of credibility, about which we conduct a delimited simulation study. In the second group, we discuss Gannon\'s modification of the p-value based on the Bayes factor and the second-generation p-value. The theory is illustrated with two case studies on pharmacotherapy in infectious diseases. Contemporary authors still refer to the p-value as a statistical indicator but have abandoned the perspective of evaluating p-values with fixed thresholds. Statistical societies worldwide should target new strategies to disseminate the debate on p-values in all applied fields of knowledge, as well as they may promote the use of different statistical procedures to supplement p-values.

The p-value is a classical proposal of statistical inference, dating back to the seminal contributions by Fisher, Neyman and E. Pearson. However, p-values have been frequently misunderstood and misused in practice, and medical research is not an exception. In recent years, in several statistical and applied journals, a debate erupted about the need of clear guidelines in reporting p-values, which culminated with the publication of the ASA statement in 2016. In this paper, we assess strengths and limitations of p-values and we assert that in applied research the p-value should be supplemented by other measures, such as the Bayes factor, the Bayes false discovery rate and the local Bayes false discovery rate. We also review a recent proposal by Bayarri et al. from a Bayesian perspective that has the advantage of introducing an indicator, the rejection odds, which keeps into account both pre- and post-experimental information, and could also have a straightforward frequentist interpretation. We conduct a delimited numerical study that investigates on the relation of the Bayes factor with its maximum, and of the local Bayes false discovery rate with its minimum under different distributional assumptions and parameter choices. We illustrate the concepts expressed in theory with an example in clinical oncology, namely a randomized trial on the effectiveness of a new chemotherapy for patients with AIDS and Kaposi\'s sarcoma.