Relation between changing the prior and the effect of an additional data point E. T. Janes writes the following in "Probability Theory: The Logic of Science":

A useful rule of a thumb is that changing the prior probability $p(\alpha | I)$ for a parameter by one power of $\alpha$ has in general about the same affect on our final conclusions as does having one more data point. This is because the likelihood function generally has a relative width $1/\sqrt{n}$, and one more power of $\alpha$ merely adds an extra small slope in the neighbourhood of the maximum, thus shifting the maximum slightly. Generally, if we have effectively $n$ independent observations, then the fractional error in an estimate that was inevitable in any event is $1/\sqrt{n}$, approximately, while the fractional change in the estimate due to one more power of $\alpha$ in the prior is about $1/n$.

I am struggling with understanding of this paragraph even though it seems quite important. It highlights that often the choice of a prior doesn't change the computations that much and hence de-emphasises debates about the appropriate choice of a prior. Yet I am struggling with the reasoning he outlines behind such proposition and I need some elaboration.
 A: So I grabbed my copy and checked what exactly he means. I think this paragraph is supposed to only refer to the specific example he is discussing in that chapter (which is, for reference, chapter 6; the example deals with determining the $n$ parameter in a binomial($\alpha$,n) distribution with known p), and I think you might be reading too much into that quote
Let's take a slightly simpler example: the good old weighted coin. We are trying to determine the parameter $\alpha$ of binomial($\alpha$,n) given data in which $m$ observations were positive. The likelihood is then:
$$ l_n(\alpha) \propto \alpha^m (1-\alpha)^{n-m} $$
In this example, we can see indeed that one additional observation (raising $n$ by 1) has a degree one contribution to the likelihood function, either multiplying it by $\alpha$ or by $(1-\alpha)$. This justifies that using a prior distribution of the form
$$ p(\alpha) \propto \alpha^{m_p} (1-\alpha)^{n_p-m_p} $$
corresponds to assuming that you have already viewed $m_p$ successes in a $n_p$ length sequence before.
However, I do not believe that this point is generally true: likelihoods very rarely contribute just a linear contribution. Here is a dumb example: imagine that you are doing the weighted coin experiment, but for some reason one observation actually corresponds to 10 new throws instead of only 1.
You can still quantify the mean contribution of an observation in terms of a Gausian with certain parameters that depend on derivatives of the log-likelihood. Look at fisher information and bernstein-von Mises theorem if you want to learn more
