## priors.minnesota_prior()


Return the Minnesota (Litterman) prior moments for VAR coefficients.


Usage

``` python
priors.minnesota_prior(
    n_lags,
    n_obs,
    tightness,
    *,
    cross_shrinkage=0.5,
    decay="harmonic",
    own_lag_mean=1.0
)
```


The prior shrinks the coefficient of variable j at lag l in the equation of variable i toward a mean that is nonzero only for the first own lag, with a standard deviation that tightens on longer lags and on cross-variable lags:

 m\_{l, ij} = \begin{cases} m\_{\text{own}} & l = 1,\\ i = j \\ 0 & \text{otherwise} \end{cases}, \qquad s\_{l, ij} = \lambda \\ d(l) \\ \begin{cases} 1 & i = j \\ \kappa & i \neq j \end{cases}, 

with d(l) = 1/l (`decay="harmonic"`) or d(l) = 1/l^2 (`decay="geometric"`, the label Impulso uses; in Doan, Litterman and Sims it is the harmonic decay with exponent two). m\_{\text{own}} = 1 is the random-walk belief for series in levels; use `own_lag_mean=0.0` for differenced or otherwise stationary series. The parameterization follows Impulso's `MinnesotaPrior` (fixed tightness rather than an estimated one, no residual-scale ratios), with `own_lag_mean` as the one addition. The classic Litterman scale ratio for the coefficient of variable j in equation i is \sigma_i / \sigma_j; apply it, when the series are on different scales, as `scale * (sigma[:, None] / sigma[None, :])`.


## Parameters


`n_lags: int`  
Number of lags p \ge 1.

`n_obs: int`  
Number of series k \ge 1.

`tightness: ArrayLike`  
Overall shrinkage \lambda \> 0 (the standard deviation of the first own lag). A modeling choice, so it has no default; a jax scalar is accepted, which lets a model sample it.

`cross_shrinkage: ArrayLike = ``0.5`  
Relative shrinkage \kappa \in \[0, 1\] of cross-variable lags versus own lags (`1.0` treats them alike, `0.0` pins them to the mean).

`decay: Literal[``"harmonic", `<span class="st">`"geometric"``]`</span>` = ``"harmonic"`  
Lag decay d(l): `"harmonic"` for 1/l, `"geometric"` for 1/l^2.

`own_lag_mean: ArrayLike = ``1.0`  
Prior mean m\_{\text{own}} of the first own lag.


## Returns


`loc: Float[Array, ``"lags obs obs"]`  
Prior means in the `phi` layout of [var_step()](var.var_step.md#numpyro_forecast.var.var_step).

`scale: Float[Array, ``"lags obs obs"]`  
Prior standard deviations in the same layout.


## Raises


`ValueError`  
If `n_lags` or `n_obs` is below one, or a Python-number `tightness` is not positive or `cross_shrinkage` is outside `[0, 1]` (an unknown `decay` is a type error under the package's runtime type checking).


## Examples

``` python
loc, scale = minnesota_prior(n_lags=2, n_obs=3, tightness=0.5, own_lag_mean=0.0)
phi = numpyro.sample("phi", dist.Normal(loc, scale).to_event(3))
```


## References

Litterman, R. B. (1986). Forecasting with Bayesian vector autoregressions: five years of experience. *Journal of Business & Economic Statistics*, 4(1).

Doan, T., Litterman, R. B. and Sims, C. A. (1984). Forecasting and conditional projection using realistic prior distributions. *Econometric Reviews*, 3(1).

Pinder, T. *Impulso*, `MinnesotaPrior` ([documentation](https://thomaspinder.github.io/Impulso/reference/generated/impulso.priors.MinnesotaPrior.html), [repository](https://github.com/thomaspinder/impulso)).