Not understand Unconstrained Representation in MCMC

Recently I am learning how to do MCMC with TFP using your Bayesian Gaussian Mixture Model example at https://github.com/tensorflow/probability/blob/master/tensorflow_probability/examples/jupyter_notebooks/Bayesian_Gaussian_Mixture_Model.ipynb.

One thing that has confused me for a long time is the purpose of having

unconstraining_bijectors = [
    tfb.SoftmaxCentered(),
    tfb.Identity(),
    tfb.Chain([
        tfb.TransformDiagonal(tfb.Softplus()),
        tfb.FillTriangular(),
    ])]

According to the descriptions mentioned above, "Hamiltonian Monte Carlo (HMC) requires the target log-probability function be differentiable with respect to its arguments. Furthermore, HMC can exhibit dramatically higher statistical efficiency if the state-space is unconstrained."

What I am not sure is this: does each element in unconstraining_bijectors refer to the bijector used for the corresponding element in initial_state?

i.e.: tfb.SoftmaxCentered() is used to transform the component weights tf.fill([components], value=np.array(1. / components, dtype), name='mix_probs'),

tfb.Identity() is used to transform the means tf.constant(np.array([[-2, -2], [0, 0], [2, 2]], dtype), as there is no need to do transform,

and tfb.Chain([tfb.TransformDiagonal(tfb.Softplus()), tfb.FillTriangular(),] is used to transform the Cholesky decomposition of the precision matrix (inversed covariance matrix) tf.eye(dims, batch_shape=[components], dtype=dtype, name='chol_precision')?

Please help me clarify. Thank you!

Start with the Bayesian_Gaussian_Mixture_Model.ipynb example linked in the issue and inspect the unconstraining_bijectors and initial_state definitions. The issue has no requested documentation change or acceptance criteria; done would require defining the clarification needed and updating the relevant explanation or example.