You use the expression ‘Go out on a Limb’ to indicate that you are supporting someone or something despite putting yourself at a potential disadvantage.
Example of use: “I’m going to go out on a limb and approve his request for additional research funding.”
The term ‘go out on a limb’ originates from the practice of climbing trees, which people have been doing since the beginning of time. The expression was first used in a figurative sense in America during the late 19th century. An early example from an October, 1895 issue of the Steubenville Daily Herald reads: “We can carry the legislature like hanging out a washing. The heft of the fight will be in Hamilton county. If we get the 14 votes of Hamilton, we’ve got ‘em out on a limb. All we’ve got to do then is shake it or saw it off.”