Sufficient Statistic and Maximum likelihood This is more a conceptual question, but it seems to me that a sufficient statistic for a parameter is a concepts that applies only if we want to estimate the parameter via maximum likelihood. Is this right or wrong?
People usually claim that sufficient statistics are a data reduction technique. That is, I can store only the summary data expressed by the form of the observed value of the sufficient statistic and I will have enough information to make inference about the parameter of interest, $\theta$, in the future. To me, this is true only if we will make inference via maximum likelihood. 
To expand, I might be able to come up with a better estimator for $\theta$ which takes on a strange functional form, and maybe then the observed value of the sufficient statistic will not be enough to allow me to infer about $\theta$ in the way that I wish. 
 A: It is not true that sufficient statistics are only relevant with maximum likelihood estimators. For instance, knowing a sufficient statistic is enough for Bayesian inference, see the calculation in Reducing dimension $P(\theta|y) = P(\theta|s)$ in the posterior distribution.

People usually claim that sufficient statistics are a data reduction technique. That is, I can store only the summary data expressed by the form of the observed value of the sufficient statistic and I will have enough information to make inference about the parameter of interest, , in the future. To me, this is true only if we will make inference via maximum likelihood.

See the Rao-Blackwell theorem: Understanding the Rao-Blackwell Theorem   and at Wikipedia.
Some other relevant posts:

*

*Maximum likelihood and sufficient statistics


*Is there a difference between Bayesian and Classical sufficiency?


*Why a sufficient statistic contains all the information needed to compute any estimate of the parameter?


*What is the correct posterior when data are sufficient statistics?
