Link adaptation (LA) optimizes the selection of modulation and coding schemes (MCS) for a stochastic wireless channel. The classical outer loop LA (OLLA) tracks the channel's signal-to-noise-and-interference ratio (SINR) based on the observed transmission outcomes. On the other hand, recent Reinforcement learning LA (RLLA) schemes sample the available MCSs to optimize the link performance objective. However, both OLLA and RLLA rely on tuning parameters that are challenging to configure. Further, OLLA optimizes for a target block error rate (BLER) that only indirectly relates to the common throughput-maximization objective, while RLLA does not fully exploit the inter-dependence between the MCSs. In this paper, we propose latent Thompson Sampling for LA (LTSLA), a RLLA scheme that does not require configuration tuning, and which fully exploits MCS inter-dependence for efficient learning. LTSLA models an SINR probability distribution for MCS selection, and refines this distribution through Bayesian updates with the transmission outcomes. LTSLA also automatically adapts to different channel fading profiles by utilizing their respective Doppler estimates. We perform simulation studies of LTSLA along with OLLA and RLLA schemes for frequency selective fading channels. Numerical results demonstrate that LTSLA improves the instantaneous link throughout by up to 50% compared to existing schemes.