I am trying to find the intuition and cons/pros of using a multivariate normal prior, gaussian random walk and independent normal distribution priors for coefficients in this context.

Lets assume i am trying to construct a regression model which generalizes w.r.t a time-attribute such as day of the month.

To do this, i constructed the following model:

y_{t, d} = \beta_0 + \beta_{t, d} * x

where as t denotes the timestep and d is an indexholder stating the day of the month, \beta_{t, d} has the following form:

Now, to my question, lets assume that we set a 4-dimensional multivariate normal prior on \beta_1, \beta_2, \beta_3, \beta_4 with a 0-mean vector and a covariancematrix filled with 1s and then we compare it with a gaussian random walk prior: \beta_1 \sim \mathcal{N}(0, 1) and \beta_t \sim \mathcal{N}(\beta_{t -1}, 1) for t = 2â€¦4. We also compare it with independently setting a prior for each \beta_t according to \beta_t \sim \mathcal{N}(0, 1).

My question is then, what can we assume about the effect on the posterior by setting these different priors. I would assume e.g that the gaussian random walk prior would result in a less wiggly and more smooth change in parameter estimates between subsequent \beta's and i would assume that the independent priors would result in the most wiggly behaviour.

Can someone shed some light onto this and further develop my thought process and secondly, if someone have seen resources doing these type of timevariant mappings priorly, please send me some links.

To make things more interesting, lets say we start to make interactions between the parameters according to the following setup for \beta_t:

Thus allowing datapoints between two dayofthemonth intervals to be a weighted minmax-like sum of â€śsubsequentâ€ť coefficients(does anyone know the name of this type of scheme?). The question stays the same ,priors and subsequent posterior dynamics are also very interesting to me in this case, i would assume that this scheme together with the GRW prior would be the setup resulting in the least wiggly behaviour thus enforcing some regularization.

I am aware that the notation might be a bit iffy, any suggestions to improve it would also be appreciated.