MCLUST model names corresponding to common models (i.e., those used for LPA / LCA) A number of reviews of mixture models, such as Fraley and Raftery (2002) describe three common models, in terms of their geometric interpretation:


*

*All mixture components are spherical and of the same size

*Equal variance

*Unconstrained variance


Helpfully, though for some beginners like me confusingly, MCLUST in R provides a wider range of model names that include the three common models above, according to the MCLUST documentation:
multivariate mixture        
"EII"   =   spherical, equal volume
"VII"   =   spherical, unequal volume
"EEI"   =   diagonal, equal volume and shape
"VEI"   =   diagonal, varying volume, equal shape
"EVI"   =   diagonal, equal volume, varying shape
"VVI"   =   diagonal, varying volume and shape
"EEE"   =   ellipsoidal, equal volume, shape, and orientation
"EVE"   =   ellipsoidal, equal volume and orientation
"VEE"   =   ellipsoidal, equal shape and orientation
"VVE"   =   ellipsoidal, equal orientation
"EEV"   =   ellipsoidal, equal volume and equal shape
"VEV"   =   ellipsoidal, equal shape
"EVV"   =   ellipsoidal, equal volume
"VVV"   =   ellipsoidal, varying volume, shape, and orientation

Which of the MCLUST model names do the three common models described by Fraley and Raftery correspond to?
My educated guesses, assuming that varying volume and shape (and orientation) are simply less-constrained parameterizations of the covariance matrix, and therefore equal volume and shape (and orientation) are the same for equal variance, are:


*

*All mixture components are spherical and the same size: EII

*Equal variance across mixture components: EEE

*Unconstrained variance across mixture components: VVV
I ask because in my area of research / field, Latent Profile Analysis (LPA) (or Latent Class Analysis [LCA]) are commonly used to do mixture modeling as part of a latent variable model approach. 
In this approach, analysts commonly specify models in which the measured variables' residual variances and covariances are constrained to be the same across profiles (or classes) or to be freely-estimated across classes. 
More generally than about this specific question, I'm searching for advice about how to interpret these geometric model descriptions to the way models are specified in an LPA / LCA approach.
 A: According to what I see in the tidyLPA package in R, they use the mclust models like this:


*

*model 1: EEI

*model 2: VVI

*model 3: EEE

*model 4: unrepresentable in mclust

*model 5: unrepresentable in mclust

*model 6: VVV
The meaning of model numbers:


*

*Equal variances, and covariances fixed to 0 (model 1)

*Varying variances and covariances fixed to 0 (model 2)

*Equal variances and equal covariances (model 3)

*Varying means, varying variances, and equal covariances (model 4)

*Varying means, equal variances, and varying covariances (model 5)

*Varying variances and varying covariances (model 6)


sources:


*

*https://github.com/data-edu/tidyLPA/blob/2c4fc51ef0d721766eb66c08d06c33e9a99dc343/R/calc_functions.R#L244

*https://github.com/data-edu/tidyLPA/blob/3a44c8654297d51c1ea74a6bc718c6206e8a83dc/vignettes/Introduction_to_tidyLPA.Rmd#L173
A: In regards to the initial posting, it is helpful to consider the geometric interpretations given above from Fraley & Rafferty/mclust documentation to the default parameterization in Mplus which invokes the "LPA" nomenclature. Note that mclust models will only be of the "LPA" variant because Guassian modeling within this package only uses continuous indicators to define the "classes" or "clusters".
The default parameterization in Mplus assumes local independence amongst profile or "class" indicators once the profile or class is taken into account. Thus: "means and variances of the indicators and the mean of the categorical latent variable are estimated as the default. The means of the latent class indicators are NOT held equal across classes as the default. The variances are held equal across classes as the default and the covariances among the latent class indicators are fixed at zero as the default” (p. 182-83 Mplus UG v8). This should be equivalent to Mclust EEI if I am not mistaken.
