# Prior selection in Gaussian - an application to height measurement

Say I have just purchased ACME's Tree Height Measuring Device (THMD). ACME states that the error $$\epsilon$$ in tree height measurement from this device can be modelled as a normal distribution with mean $$\mu$$=0 and variance $$\sigma$$2, so for any one measurement of a tree with height h, the reading from the THMD will be height = h + $$\epsilon$$ and the height of a tree as measured by the THMD can be modelled as a normal distribution with $$\mu$$h=h and variance=$$\sigma$$h2=$$\sigma$$2

Say I use this THMD to take two measurements of a tree and get a values of h1 and h2. I want to know the height of my tree with as little uncertainty as possible. One way I can think of to do this is Bayesian updating of a normal distribution with new data. As I take more and more measurements, I can continue to update the estimated value of $$\mu$$h and reduce the uncertainty in this estimate. The relevant equations for updated mean and variance of the mean from the above link are pasted below for convenience.

I'm struggling with deciding how set up the problem. My first inclination is to say that the first measurement h1 is the best estimate I have of $$\mu$$h and to use this as the prior $$\mu$$0. I am also inclined to use the height variance from the tool as the prior variance $$\sigma$$02=$$\sigma$$h2=$$\sigma$$2. This is because with just h1, I have one measurement of height, and the standard error of the mean is $$\sigma$$h/$$\sqrt{n}$$ where n is 1. I then use equations 20 and 24 from my linked source above to get the updated values of $$\mu$$n and $$\sigma$$n using $$\overline{x}$$=h2 and n=1. As it turns out, it seems this is equivalent to just invoking the law of large numbers, where the best guess for $$\mu$$h is simply the average of my two measurements and the variance of this value is $$\sigma$$2/n.

Another approach is to just use arbitrary values for priors where $$\mu$$0=0 and $$\sigma$$0=1 and then updating these values twice; once for each measurement.

I have little experience in Bayesian updating and prior selecting. My first approach seems to be reasonable, but I would appreciate any feedback anyone is willing to provide.