Prior Distributions

There are two ways to specify prior distributions in blavaan. First, each type of model parameter has a default prior distribution that may or may not be suitable for your specific situation. You are free to modify the defaults. Second, the priors for individual model parameters can be specified in the model syntax. Each is discussed below.

Defaults

The default priors can be seen via

dpriors()
               nu             alpha            lambda              beta 
   "normal(0,32)"    "normal(0,10)"    "normal(0,10)"    "normal(0,10)" 
            theta               psi               rho             ibpsi 
"gamma(1,.5)[sd]" "gamma(1,.5)[sd]"       "beta(1,1)" "wishart(3,iden)" 
              tau             delta 
"normal(0,10^.5)" "gamma(1,.5)[sd]" 

It is important to note that these prior distributions correspond to Stan parameterizations. These are similar to R parameterizations but not necessarily exactly the same. The Greek(ish) names above correspond to the following parameter types (where MV is manifest/observed variable and LV is latent variable):

                 nu               alpha              lambda                beta 
     "MV intercept"      "LV intercept"           "Loading"        "Regression" 
              theta                 psi                 rho               ibpsi 
     "MV precision"      "LV precision"       "Correlation" "Covariance matrix" 

For target = "stan" (the default), priors are currently restricted to one distribution per parameter type. You can change the prior distribution parameters (for example, the mean and standard deviation of a normal), but you cannot change the prior distribution type. More flexibility will be added in the future. If you require more flexibility right now, change the target to either "stanclassic" (the old Stan approach) or "jags" (the JAGS approach).

To modify prior distributions, we could simply supply a new text string to dpriors() like this:

mydp <- dpriors(lambda="normal(1,2)")

so that the default prior for loadings is now normal with mean 1 and standard deviation 2, and the rest of the parameters remain at the original defaults. Any univariate distribution defined in Stan could be used for the prior here. The next time we estimate a model (via bsem(), bcfa(), or blavaan()), we would add the argument dp=mydp to use this new set of default priors.

Individual Parameters

In addition to setting the prior for one type of model parameter, the user may wish to set the prior of a specific model parameter. This is accomplished by using the prior() modifier within the model specification. For example, consider the following syntax for the Holzinger & Swineford (1939) confirmatory factor model:

HS.model <- ' visual  =~ x1 + prior("normal(1,2)")*x2 + x3
              textual =~ x4 + x5 + prior("exponential(1)")*x6
              speed   =~ x7 + x8 + x9 
              x1 ~~ prior("gamma(3,3)[sd]")*x1 '

The loading from visual to x2 now has a normal prior with mean 1 and standard deviation 2, while the loading from textual to x6 has an exponential prior with rate 1. All other loadings have the default prior distribution.

In the above syntax, we have additionally specified a gamma(3,3) prior associated with the residual of x1. The [sd] text at the end of the distribution says that this prior goes on the residual standard deviation, as opposed to the residual precision or residual variance. For target="stanclassic" or target="jags", there exists two more options: a [var] option for the residual variance, and no brackets for the precision. This bracketed text can be used for any model variance/precision parameter and could also be used in default prior specification if desired.

Covariance Parameters

One additional note on covariance parameters defined in the model syntax: the prior() syntax specifies a prior on the correlation associated with the covariance parameter, as opposed to the covariance itself. The specified distribution should have support on (0,1), and blavaan automatically translates the prior to an equivalent distribution with support on (-1,1). It is safest to stick with beta priors here. For example, the syntax

HS.model <- ' visual  =~ x1 + x2 + x3
              textual =~ x4 + x5 + x6
              speed   =~ x7 + x8 + x9 
              visual ~~ prior("beta(1,1)")*textual '

places a Beta(1,1) (uniform) prior on the correlation between the visual and textual factors. If desired, we could also specify priors on the standard deviations (or variances or precisions) of the visual and textual factors. Together with the prior on the correlation, these priors would imply a prior on the covariance between visual and textual.