Finding the shape and scale of a gamma distribution given home range

Question

pasted below is supplemental material from an article. After finding home range values, the authors say, "gamma(13,10) covers this nicely". How did the authors find 13 and 10?

# Moldenhauer and Regelski (1996) state that home range size typically
# ranges from 0.08-0.65 ha, which equates to a radius of
sqrt(0.08/pi*10000) # 15.96 m or
sqrt(0.65/pi*10000) # 45.49 m

# Since parulas were detected by song, let's be safe
# and add a minimum and maximum
# distance at which they could be heard, 50 and 250 m, based upon
# Simons et al. (2009).
# So now we have an area between
(15.96+50)^2*pi / 10000  # 1.37 ha, and
(45.49+250)^2*pi / 10000 # 27.4 ha

# We note that this is a very large range.

# Following Royle et. al (2011), and assuming a
# chi-squared distribution with 2 degrees of freedom,
# the range of sigma is given by
sqrt(1.37*10000/pi)/sqrt(5.99)   # 27 m
sqrt(27.4*10000/pi)/sqrt(5.99)   # 120 m

# In our grid spacing, 1 unit = 50m, so our we want a prior with most
# of the density between:
27/50  # 0.54
121/50 # 2.42

# Gamma(13, 10) covers this nicely
qgamma(c(0.001, 0.5, 0.999), 13, 10)

plot(function(x) dgamma(x, 13, 10), 0, 5, xlim=c(0, 3), ylim=c(0, 1.5))**

Answer 1

Honestly, I think the answer to this may just be a bunch of hand waving. If I were to choose a gamma prior and had some intuition about the range I would find the median value and make it the mean of the gamma distribution. So for this example I would do something like:

# Find median of range (which is 1.48)
gamma_med <- median(seq(0.54, 2.42, length.out = 1e6))

The gamma distribution has two parameters, gamma(a,b). The first moment, or the mean, can be calculated easily because it is just a/b. Therefore, if we want our mean to be 1.48 we just need to choose a shape (a) and scale(b) whose ratio equals 1.48. The simplest one would be gamma(14.8, 10), but we could increase or decrease our variance by changing those parameters (variance of gamma = a/(b^2)).

Here is how different the two priors are, you can see how gamma(13,10) is more precise. At the end of the day though it really comes down to what you feel is defensible in what you want your priors to be (or preferably you should use multiple priors to see how they influence your posterior).