You are entitled to be free on bond under the Michigan Constitution Art 1, § 15. Drunk driving arrests are permissible warrantless arrests, but the accused must be arraigned as soon as possible. This must be done without any undue delay, even if the charge is a felony drunk driving charge. The United States Supreme Court in County of Riverside v McLaughlin, 500 US 44 (1991), held that an arrest without a warrant is unreasonable if it is not followed by arraignment within forty-eight hours.
Warrantless arrests are generally unlawful, so there are special rules under the law that deal with warrantless arrests. The police cannot simply keep you locked up on their good word. They must either make an interim bond available, release you without charges, or get you in front of a judge ASAP. Keeping you in custody too long exposes the police department to civil liability.