The answer provided by Henry highlights some fascinating statistical theory; for another pertinent theoretical link, see wikepedia's article Categorical distribution.
Here we want to proceed pragmatically, but in the end the inquisitive student of statistics should be motivated to study the theoretical underpinnings.
OK, how much deviation do we think is possible? Now $\frac{1}{6} = 0.1666...$ and
$\frac{2}{7} = 0.2857...$
$\frac{3}{13} = 0.2307...$
$\frac{4}{19} = 0.21052...$
Stop! We don't have that much confidence in our die - if a face came up $21$% of the time we would not be surprised.
OK, now imagine throwing a die $18$ times and each side comes up exactly $3$ times. On the $19$ throw whatever comes up would be assigned the statistical frequency of $\frac{4}{19}$.
So there you have the simple idea of how to update the discrete uniform random variable as each data outcome is observed. You make believe the first real observed point is the $19$th reading, the second real reading is the $20$th, etc.; where the first eighteen are imagined to comes up showing an occurrence of $3$ for each of the six sides.
Example: The first (real) row of the die shows a $2$. Updates:
$p(1) = \frac{3}{19}$
$p(2) = \frac{4}{19}$
$p(3) = \frac{3}{19}$
$p(4) = \frac{3}{19}$
$p(5) = \frac{3}{19}$
$p(6) = \frac{3}{19}$