Inference about parameter $\theta$ be same? Let $\mathbf x$ be a sample point and $T(\mathbf x)$ be a statistic of $\mathbf x$.
Similarly, let $\mathbf y$ be a sample point and $T(\mathbf y)$ be a statistic of $\mathbf y$.
In the book Statistical Inference by George Casella, it is written that :
If $\mathbf x$  and $\mathbf y$ are two sample points such that $T(\mathbf x)=T(\mathbf y)$, then the inference about the parameter $\theta$ would be the same.
I have not understood the statement. $T(\mathbf x)= T(\mathbf y)$ does not imply $\mathbf x= \mathbf y$. Then why should the inference about parameter $\theta$ be same ?
 A: This is only true when $T$ is a sufficient statistic.  
Sufficiency implies that knowing the value of the statistic provides just as much information as having the original data points, given some assumption about how the data was generated.  (So a statistic may be sufficient for some assumed data generating processes but insufficient for others).  
Consider 5 independent flips of the same coin to learn about its bias.  Making a Binomial assumption about this data implies that only the number of heads and tails are relevant for inference about the proportion parameter representing the true bias of the coin.  That's because any ordering of 3 heads and 2 tails imply the same likelihood function, and therefore the same maximum and curvature.  That's what makes these statistics sufficient.
See this page for some more discussion.
A: Because all inference we do is based on information we got from statistics, and since they are the same the inference must be the same.
A: This is not true in general.
This is true if and only if $T$ is a sufficient statistic for $\theta$ - (by definition)
Actually, your citation comes from the definition of Sufficiency Principle from that book
