Bayesian One Sample Inference: Normal Priors

You can specify the following prior distribution criteria for your Bayesian One-Sample Inference:

Note: Many applied researchers may question the need to specify a prior. Reference priors minimize the concern where the prior is generally overwhelmed as the data increases. When informative prior information is specified, Bayesian methods can effectively use the information. The requirement to specify a prior should not be viewed as a deterrent to using Bayesian analysis.

Prior on Variance/Precision

Provides options for defining variance and precision values.

Variance

Select to specify the prior distribution for the variance parameter. When this option is selected, the Prior Distribution list provides the following options:

Note: When the data variance is already specified for some variables, the following settings are ignored for those variables.

Diffuse - the default setting. Specifies the diffuse prior.
Inverse Chi-Square - Specifies the distribution and parameters for inverse-χ²(ν₀,σ²₀), where ν₀ > 0 is the degree of freedom, and σ²₀ > 0 is the scale parameter.
Inverse Gamma - Specifies the distribution and parameters for inverse-Gamma(α₀, β₀), where α₀> 0 is the shape parameter, and β₀ > 0 is the scale parameter.
Jeffreys S2 - Specifies the non-informative prior ∝ 1/σ²₀.
0.
Jeffreys S4 - Specifies the non-informative prior ∝ 1/σ⁴₀.

Precision

Select to specify the prior distribution for the precision parameter. When this option is selected, the Prior Distribution list provides the following options:

Gamma - Specifies the distribution and parameters for Gamma (α₀, β₀), where α₀ > 0 is the shape parameter, and β₀ > 0 is the scale parameter.
Chi-Square - Specifies the distribution and parameters for χ²(ν₀), where ν₀ > 0 is the degree of freedom.

Shape Parameter: Specify the shape parameter a₀ for Inverse-Gamma distribution. You must enter a single value that is greater than 0.
Scale Parameter: Specify the scale parameter b₀ for Inverse-Gamma distribution. You must enter a single value that is greater than 0. The larger the scale parameter, the more spread out the distribution.

Prior on Mean Given Variance/Precision

Specify the prior distribution for the mean parameter that is conditional on the variance or the precision parameter.

Normal

Specifies the distribution and parameters for Normal(μ₀, K^-1₀σ²₀) on variance or Normal(μ₀, K₀/σ²₀) on precision, where μ₀∈ (-∞, ∞) and σ² > 0.

Location Parameter: Enter a numeric value that specifies the location parameter for the distribution.
Scale Parameter: Specify the scale parameter b₀ for Inverse-Gamma distribution. You must enter a single value that is greater than 0.
Kappa: Specify the value of K₀ in Normal(μ₀, K^-1₀σ²₀) or Normal(μ₀, K₀/σ²₀). You must enter a single value that is greater than 0 (1 is the default value).

Diffuse

The default setting that specifies the diffuse prior ∝ 1.