Born in 1837. The son of a priest. In his youth he served in the Danish army. In 1864 he took part in the Danish War. He rose to the rank of lieutenant. After leaving military service, he worked as a teacher, translator and journalist. Participated in the movement for women’s equality, in 1871 he co-founded the Danish Women’s Association. From 1872 he was elected to parliament from the left liberals. Bayer was a supporter of close cooperation between the countries of Scandinavia and Denmark, an activist of the ideas of neutrality.
In 1882 he founded the Association for Danish Neutrality (transformed in 1885 into the Danish Peace Association). He was delegated to the first Scandinavian world congress (1885), and since 1889 – to world congresses.
In 1891, Bayer co-founded the International Peace Bureau, headquartered in Bern. During 1891-1907 he was chairman and honorary president of the bureau.
For his efforts aimed at the peaceful solution of the problem of the Norwegian-Swedish union (union), together with the famous public figure of Sweden, Klas Pontus Arnoldson, he was awarded the Nobel Peace Prize in 1908.
Died January 22, 1922