a function 𝑓(𝐷 = {𝑋₁, ..., 𝑋_𝑛}) is a sufficient statistic is a type of statistic where the following cases are satisfied:

• 𝑓 is a function that maps instances 𝐷 = {𝑋₁, ..., 𝑋_𝑛} → vector ℝ^𝑘
• for any 2 datasets: 𝐷 and 𝐷', and any 𝜃∊𝚯 we have:
  • 𝑓(𝐷) = 𝑓(𝐷') ⇒ 𝐿(𝜃|𝐷) = 𝐿(𝜃|𝐷')

Sufficient Statistic - Example

Given:

• data 𝐷 = {𝑋₁, ..., 𝑋_𝑛} where each 𝑋_𝑖 is sampled from a Bernoulli Distribution

The possible examples of sufficient statistics 𝑓(𝐷) for the likelihood of Bernoulli Distribution are:

• one that outputs the number of 1's in the data 𝐷
• one that outputs the number of 0's in the data 𝐷