  The available liquidity at any time in financial markets falls largely short
of the typical size of the orders that institutional investors would trade. In
order to reduce the impact on prices due to the execution of large orders,
traders in financial markets split large orders into a series of smaller ones,
which are executed sequentially. The resulting sequence of trades is called a
meta-order. Empirical studies have revealed a non-trivial set of statistical
laws on how meta-orders affect prices, which include i) the square-root
behaviour of the expected price variation with the total volume traded, ii) its
crossover to a linear regime for small volumes, and iii) a reversion of average
prices towards its initial value, after the sequence of trades is over. Here we
recover this phenomenology within a minimal theoretical framework where the
market sets prices by incorporating all information on the direction and speed
of trade of the meta-order in a Bayesian manner. The simplicity of this
derivation lends further support to the robustness and universality of market
impact laws. In particular, it suggests that the square-root impact law
originates from the over-estimation of order flows originating from
meta-orders.
