Suppose I have some random process $X$ which is emitting values which follow a normal distribution:
$$X \sim N(μ, σ^2)$$
Both $μ$ and $σ$ are unknown, so I want to model each of them with their own distribution which I will update every time I observe a new value.
How can I do this?
For $μ$ it seems obvious that I should model it with its own normal distribution: $μ \sim N(μ_μ, σ_μ^2)$. For $σ^2$ it's not clear what distribution I should use - my googling so far suggests that inverse-gamma would make the math work-out nicely but it's not clear to me that it even makes sense to use two independent distributions for $μ$ and $σ^2$.
So my question is: what mathematical model should someone use in this situation (or, if there's a choice, what are the options), and how exactly does one calculate the posterior parameters of the model given the prior parameters and an observation $x$?