We discuss scalar-tensor theories based on a non-Riemannian geometry, called the metric-affine geometry, where the metric and the connection are treated as independent variables. In the metric-affine formalism, the Einstein-Hilbert action enjoys an additional local symmetry, the projective symmetry, under a shift of the connection. We find that the projective symmetry can provide an Ostrogradky ghost-free structure of general scalar-tensor theories. The ghostly sector of the second-order derivative of the scalar is absorbed into the projective gauge mode when the unitary gauge can be imposed. We also find that, up to the quadratic order of the connection, the most general projective invariant theory is equivalent to the U-degenerate theory when the connection is integrated out and, if we further assume the Galileon-type self-interactions, the theory is equivalent to DHOST theory.
 Katsuki Aoki and Keigo Shimada, Galileon and generalized Galileon with projective invariance in a metric-affine formalism, 1806.02589.
 Katsuki Aoki and Keigo Shimada, Scalar-metric-affine theories: Can we get ghost-free theories from symmetry?, 1904.10175.