x’ forecast only on HP-32E

[Nigel (UK)]

(06-28-2023 01:37 PM)rawi Wrote:  Hi Nigel,

in your equation, if Epsilon has a normal distribution with mean zero and standard deviation s your x has as well a normal distribution with mean zero and standard deviation s/b. The point is that x is not a random variable and does not have a distribution. That is why we minimize in regression analysis the sum of squares in the direction of the y axis. If x and y both have a distributuion you need another model and you will get a different beta. You can read about this here: https://en.wikipedia.org/wiki/Errors-in-...les_models

Although x may not be a random variable, my knowledge of its value (given a definite value of y) is uncertain. I take the Bayesian view that the probability distribution for such a variable is a reflection of this uncertainty in my knowledge; indeed, this (or something close) is the Bayesian definition of probability. (Heresy? Surely not!)

I agree that the equation in the second paragraph of my previous post is unlikely to give the correct (Bayesian) answer - using precise values for a and b will underestimate the uncertainty in x - but the approach in the third paragraph should work, I think.

As I said, I am not remotely an expert on this subject!

Nigel (UK)