Simple upper and lower bounds for the multivariate Laplace approximation

We propose a new proof of the Laplace approximation for multiple integrals and consequently find new bounds for the approximation error. The main advantages of our approach are its simplicity, an explicit form of the bounds and small coefficient in the main error term, which depend on the smallest eigenvalue of the Hessian matrix, only. On the other hand, the method seems indicating much larger values of the parameter then really necessary.