Bayesian approximations and extensions: optimal decisions for small brains and possibly big ones too.

We compared the performance of Bayesian learning strategies and approximations to such strategies, which are far less computationally demanding, in a setting requiring individuals to make binary decisions based on experience. Extending Bayesian updating schemes, we compared the different strategies while allowing for various implementations of memory and knowledge about the environment. The dynamics of the observable variables was modeled through basic probability distributions and convolution. This theoretical framework was applied to the problem of male fruit flies who have to decide which females they should court. Computer simulations indicated that, for most parameter values, approximations to the Bayesian strategy performed as well as the full Bayesian one. The linear approximation, reminiscent of the linear operator, was notably successful, and, without innate knowledge, the only successful learning strategy. Besides being less demanding in computation and thus realistic for small brains, the linear approximation was also successful at limited memory, which would translate into robustness in rapidly changing environments. Knowledge about the environment boosted the performance of the various learning strategies with maximal performance at large utilization of memory. Only for limited memory capacities, intermediate knowledge was most successful. We conclude that many animals may rely on algorithms that involve approximations rather than full Bayesian calculations because such approximations achieve high levels of performance with only a fraction of the computational requirements, in particular for extensions of Bayesian updating schemes, which can represent universal and realistic environments.