Improve default GaussianRandomWalk and AR init distributions

[ricardoV94]

When refactoring the GaussianRandomWalk to V4, we set the init to default to Normal(mu, sigma), which does not make much sense. mu is the drift parameter, and says nothing about the expected starting location of the series. sigma probably also makes little sense, as it assumes the uncertainty about the initial location has anything to do with the innovations noise.

Possible alternatives:

• init = Normal.dist(0, 1)
• init = Flat.dist(), which was the default in V3, but it means that prior and posterior predictive will fail by default.

We should also rename init to init_dist to emphasize this parameter must de a distribution.

pymc/pymc/distributions/timeseries.py

Line 239 in c76b9b9

init = Normal.dist(mu, sigma)

A similar question arises in the AR, where we again reuse the innovation noise sigma for the uncertainty about the initial location of the series.

pymc/pymc/distributions/timeseries.py

Line 429 in c76b9b9

init_dist = Normal.dist(sigma=at.shape_padright(sigma), size=(*sigma.shape, ar_order))

[ricardoV94]

Reopening the issue @twiecki.

After #5779 we are using unit normals as default. Other proposed options:

1. Use Flat again and let it fail when doing predictive sampling
2. Use a unit normal with larger sigma
3. Do not provide a default, user must always specify it.

[twiecki]

Decision: Raise warning if not specified, increase standard deviation to 100.

[AlexAndorra]

I'll take this on

[lucianopaz]

Is it Possible to set it to the stationary distribution? At least for the AR?

[lucianopaz]

Another alternative is to set it to a Dirac delta at 0

[ricardoV94]

@lucianopaz I think the initial point does not affect the stationarity?

[lucianopaz]

@lucianopaz I think the initial point does not affect the stationarity?

I meant to set the initial distribution to the AR stationary distribution. For AR1 there’s a formula in terms of the coefficients. I don’t know what happens for AR(p) though

[ricardoV94]

@lucianopaz I think the initial point does not affect the stationarity?

I meant to set the initial distribution to the AR stationary distribution. For AR1 there’s a formula in terms of the coefficients. I don’t know what happens for AR(p) though

Won't that perhaps introduce very strong constraints on the initial point?

Btw, a DiracDelta would break everything, as no draws would ever be accepted unless we implemented a custom sampler (or transform) for the AR alone