Note: I'm a stats novice, please let me know if any of these terms are unclear or misused and I'll update the question!
I'd like to predict future values of a time series. More precisely, I'm interested in the distribution of the next value.
I've chosen a normal/Gaussian prior. I see that I can calculate my new posterior distribution by solving the conjugate prior (https://en.wikipedia.org/wiki/Conjugate_prior, Continuous Distributions section, normal with known variance).
This seems to treat all observations of my time series equally. What if my time series has an underlying trend upward or down - is there a way to represent and quantify this?
For example, I've taken an existing time series with ~30 data points in the range of 30-50. I now add 10 fake data points that are double the max observed value. This barely moves my distribution's mean by ~4. This is across a range of mean/variance combinations for my prior. With 25% of my observations so far above the prior's mean, and especially because they all occur recently, my intuition is that I should see a large shift in posterior mean and posterior variance.
Is my intuition wrong, or is there something like a "Bayes moving normal" that I can use?