A spin system can be thought of as an information coding system that transfers information of the interaction configuration into information of the equilibrium state of the spin variables. Hence it can be expected that the relations between the interaction configuration and equilibrium states are consistent with the known laws of information theory. We show that Shannon's rate-distortion theorem can be used to obtain an universal constraint on neighboring spin correlations for a broad range of Ising spin systems with two-body spin interactions. Remarkably, this constraint gives a bound for the multicritical point in the phase diagram, when a mean-field behavior for the neighboring spin pairs can be expected in the paramagnetic phase.