# Sufficient statistics for layman

Can someone please explain sufficient statistics in very basic terms? I come from an engineering background, and I have gone through a lot of stuff but failed to find an intuitive explanation.

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Here's one example. Suppose $X$ has a symmetric distribution about zero. Instead of giving you a sample, I hand you a sample of absolute values instead (that's the statistic). You don't get to see the sign. But you know that the distribution is symmetric, so for a given value $x$, $-x$ and $x$ are equally likely (the conditional probability is $0.5$). So you can flip a fair coin. If it comes up heads, make that $x$ negative. If tails, make it positive. This gives you a sample from $X'$, which has the same distribution as the original data $X$. You basically were able to reconstruct the data from the statistic. That's what makes it sufficient.
In Bayesian terms, you have some observable property $X$ and a parameter $\Theta$. The joint distribution for $X,\Theta$ is specified, but factored as the conditional distribution of $X\mid \Theta$ and the prior distribution of $\Theta$. A statistic $T$ is sufficient for this model if and only if the posterior distribution of $\Theta\mid X$ is the same as that of $\Theta\mid T(X)$, for every prior distribution of $\Theta$. In words, your updated opinion about $\Theta$ after knowing the value of $X$ is the same as your updated opinion about $\Theta$ after knowing the value of $T(X)$, whatever prior opinion you have about $\Theta$. Keep in mind that sufficiency is a model dependent concept.