Distributional priors are not set correctly for models with heteroskedasticity

[ivlis]

This works as expected

df = pd.DataFrame({'x':[1,2,3], 'y' : [3,2,1]})

m1 = bmb.Model(
    bmb.Formula(
        "x ~ y",
    ),
    data=df,
    family="negativebinomial",
    priors={"alpha":
         bmb.Prior("InverseGamma", alpha=0.2, beta=0.3),
    },
)

what results into a model:

       Formula: x ~ y
        Family: negativebinomial
          Link: mu = log
  Observations: 3
        Priors: 
    target = mu
        Common-level effects
            Intercept ~ Normal(mu: 0.0, sigma: 6.6144)
            y ~ Normal(mu: 0.0, sigma: 3.0619)
        
        Auxiliary parameters
            alpha ~ InverseGamma(alpha: 0.2, beta: 0.3)

However if I specify a formula on alpha, setting priors does not work

    bmb.Formula(
        "x ~ y",
        "alpha ~ y",
    ),
    data=df,
    family="negativebinomial",
    priors={
        "alpha_y": bmb.Prior("InverseGamma", alpha=0.2, beta=0.3),
        "alpha_Intercept": bmb.Prior("InverseGamma", alpha=0.2, beta=0.3),
    },
)

this results into default priors:

       Formula: x ~ y
                alpha ~ y
        Family: negativebinomial
          Link: mu = log
                alpha = log
  Observations: 3
        Priors: 
    target = mu
        Common-level effects
            Intercept ~ Normal(mu: 0.0, sigma: 6.6144)
            y ~ Normal(mu: 0.0, sigma: 3.0619)
    target = alpha
        Common-level effects
            alpha_Intercept ~ Normal(mu: 0.0, sigma: 1.0)
            alpha_y ~ Normal(mu: 0.0, sigma: 1.0)

I tried - set_prior(), it gives the same result.

It could be that I'm doing something not correctly, but models with multiple targeted are not documented very well, imo.

[tomicapretto]

@ivlis you just need to pass a nested dictionary. It works this way to not have Bambi guessing the name of variables after a _.

m1 = bmb.Model(
    bmb.Formula(
        "x ~ y",
        "alpha ~ y",
    ),
    data=df,
    family="negativebinomial",
    priors={
        "alpha": {
            "y": bmb.Prior("InverseGamma", alpha=0.2, beta=0.3),
            "Intercept": bmb.Prior("InverseGamma", alpha=0.2, beta=0.3)
        }
    },
)

m1

       Formula: x ~ y
                alpha ~ y
        Family: negativebinomial
          Link: mu = log
                alpha = log
  Observations: 3
        Priors: 
    target = mu
        Common-level effects
            Intercept ~ Normal(mu: 0.0, sigma: 6.6144)
            y ~ Normal(mu: 0.0, sigma: 3.0619)
    target = alpha
        Common-level effects
            alpha_Intercept ~ InverseGamma(alpha: 0.2, beta: 0.3)
            alpha_y ~ InverseGamma(alpha: 0.2, beta: 0.3)

And if you need to set priors for the linear prediction of mu you can do

bmb.Model(
    bmb.Formula(
        "x ~ y",
        "alpha ~ y",
    ),
    data=df,
    family="negativebinomial",
    priors={
        "mu": {
            "Intercept": bmb.Prior("Normal", mu=0, sigma=1),
            "y": bmb.Prior("Normal", mu=0, sigma=0.5),
        },
        "alpha": {
            "y": bmb.Prior("InverseGamma", alpha=0.2, beta=0.3),
            "Intercept": bmb.Prior("InverseGamma", alpha=0.2, beta=0.3)
        }
    },
)

       Formula: x ~ y
                alpha ~ y
        Family: negativebinomial
          Link: mu = log
                alpha = log
  Observations: 3
        Priors: 
    target = mu
        Common-level effects
            Intercept ~ Normal(mu: 0.0, sigma: 1.0)
            y ~ Normal(mu: 0.0, sigma: 0.5)
    target = alpha
        Common-level effects
            alpha_Intercept ~ InverseGamma(alpha: 0.2, beta: 0.3)
            alpha_y ~ InverseGamma(alpha: 0.2, beta: 0.3)

[ivlis]

Oh, great, thank you! I wonder if this can be better documented. The doc page mentions nothing about this:

Optional specification of priors for one or more terms. A dictionary where the keys are the names of terms in the model, “common,” or “group_specific” and the values are instances of class Prior. If priors are unset, uses automatic priors inspired by the R rstanarm library.

https://bambinos.github.io/bambi/api/Model.html#bambi.Model

I can create a PR.

[tomicapretto]

@ivlis it would be amazing, thanks!