Consider a Poisson point process with a Gamma prior. The answer here takes a shape-rate parametrization for the Gamma (interpreted as $a$ exponential distributions each with rate $b$ added together). If we observe $X_1$, $X_2$, $\dots$, $X_n$ events in one unit of time each, we get the posterior which is also gamma distribution with updated shape parameter:
$$a'=a+\sum_i X_i$$
and rate:
$$b'=b+n$$
The updated shape makes sense since we have now observed $a+\sum_i X_i$ exponential distributions. But I can't figure out how the updated rate parameter makes sense. The added $n$ can be interpreted as the number of additional events or the amount of additional time. Either way, adding to a rate which has units of inverse time doesn't make sense. What am I missing?
The current estimate for the rate $\lambda$ of the Poisson point process is $\frac{a'}{b'}$. If this is to be represented by a sum of $a'$ exponentially distributed variables, each must have mean $\frac1{b'}$. This is the scale parameter of the gamma distribution, and the rate parameter is its inverse.
The rate parameter doesn’t have units of inverse time. It has units of time, since it’s the reciprocal of a parameter proportional to the mean of the rate $\lambda$, which has units of inverse time.