# Basic intuition about minimal sufficient statistic

As stated by Wikipedia:

A sufficient statistic is minimal sufficient if it can be represented as a function of any other sufficient statistic. In other words, $$S(X)$$ is minimal sufficient if and only if $$S(X)$$ is sufficient, and if $$T(X)$$ is sufficient, then there exists a function f such that $$S(X) = f(T(X))$$. Intuitively, a minimal sufficient statistic most efficiently captures all possible information about the parameter $$\theta$$.

I have some trouble understanding the full meaning of "minimal"

What I don't get is, what happens if we have a third sufficient statistic, that we call $$U(X)$$ such that:

$$U(X)=g(S(X))=f(g(T(X)))$$

In this case are both $$U(X)$$ and $$S(X)$$ minimal? I ask this because minimal makes me think that it must be the "most minimal" so there can be only one (group) of minimal statistics.

If I am not wrong if f or g are invertible Then the three sufficient statistics are all minimal but in this case they all belong to the same group. In the case f and g are not invertible,the three statistics: $$U(X),S(X),T(X)$$ have all different efficiency in capturing information

Let the sample space be $$\mathcal{X}$$. Then a sufficient statistic $$T$$ can be seen as indexing a partition of $$\mathcal{X}$$, that is, $$T(x)=T(y)$$ iff (if and only if) $$x,y$$ belongs to the same element of the partition. A minimallly sufficient statistic is then giving a maximal reduction of the data. That is to say, if $$T$$ is minimally sufficient, then if we take the partition corresponding to $$T$$, take two distinct elements of that partition, and makes a new partition by replacing the two by their union, the resulting statistic is not longer sufficient. So, any other sufficient statistic, say $$S$$, which is not minimal, will have a partition which corresponds to a refinement of the partition of $$T$$, that is, every element of the partition of $$T$$ is a union of elements of the partition of $$S$$ (this becomes easier to understand if you make a drawing from my text!). So, when you know the value of $$S$$, you know in which element of the partition of $$S$$ that sample point belongs, and also in which element of the partition of $$T$$ that sample point belongs — since that partition is coarser. That is what it means when it says that $$T$$ is a function of every other sufficient statistic — every other sufficient statistic gives more information (or the same information) about the sample than what $$T$$ does.
Definition: A partition of $$\mathcal{X}$$ is a collection of subsets of $$\mathcal{X}$$ such that $$\cup_{\alpha} \mathcal{X}_\alpha = \mathcal{X}$$ and
$$\mathcal{X}_\alpha \cap \mathcal{X}_\beta = \emptyset$$ unless the two elements of the partition are identical, that is , $$\alpha=\beta$$.