# Why marginal impact of an observation to the posterior decreases in Bayesian inference?

When we toss a coin with an unknown Heads probability $$p$$, we can use Bayesian inference to estimate the unknown value $$p$$.

Say, we start with a Beta prior distribution with parameters $$(a,b)$$ and then update the prior as we observe Heads or Tails.

At the point we tossed $$n$$ times and have observed $$n_H$$ number of Heads, we say the probability of Head is $$\hat p_H=\frac{a+n_H}{a+b+n}=\frac{a}{a+b+n}+\frac{n_H}{a+b+n}.$$

Here, as we can see, if $$n_H$$ is increases by one, the estimate $$\hat p_H$$ increase by $$1/(a+b+n)$$.

I wonder if there is an intuitive explanation of why this increment of the estimate decreases as $$n$$ increases. Any idea?

Hence, if $$n$$ increases, the contribution of any particular "head" to the total estimate decreases. When you move from $$n=1$$ to $$n=2$$ the first observation still is pretty important to form the posterior mean, while if $$n=1,000$$, any particular observation is relatively less influential.