> [!definition] > > The maximum likelihood rule is a [[Decision Rule|decision rule]] that assigns each received [[Code|codeword]] the most likely sent codeword. > > Let $C = \list{x}{N}$ be the [[Set|set]] of all possible codewords, and $y$ be the codeword received. Consider the $N$ [[Conditional Probability|conditional probabilities]] $P_{y}(x_j\ \text{sent})$. We assign $x_j$ to be the sent codeword if > $ > P_y(x_j\ \text{sent}) \gt P_y(x_k\ \text{sent}) \quad \forall j \ne k > $ > > If the code is optimal and the [[Probability Distribution|distribution]] of the codewords is [[Uniform Distribution|uniform]], then by [[Bayes' Theorem]], > $ > \begin{align*} > \frac{P(x_j\ \text{sent})P_{x_j}(y)}{P(y)} &\gt \frac{P(x_k\ \text{sent})P_{x_k}(y)}{P(y)} \quad \forall j \ne k \\ > P_{x_j}(y) &\gt P_{x_k}(y) \quad \forall j \ne k > \end{align*} > $ > > If multiple codewords have the same maximum probability, then one will be chosen at random.