The reason we do not call ( p(y | z, c) ) the posterior is because, in Bayesian terms, posterior refers specifically to the distribution of the unknown (latent) variable ( z ) after observing some data ( y ) and any known input ( c ).

Here's a more detailed explanation:

1. What is ( p(y | z, c) )?

( p(y | z, c) ) is the likelihood. It expresses the probability of observing the output ( y ) (the data) given the latent variable ( z ) and the input ( c ).

• Likelihood: In Bayesian terms, the likelihood refers to how probable the observed data ( y ) is, given specific values of the latent variable ( z ) and any known inputs ( c ). The likelihood helps update our belief about the unknown variable ( z ), but it is not itself a "belief" about ( z ).

[ p(y | z, c) = y z c ]

Why it's not called the posterior: The likelihood is conditional on the latent variable ( z ) and the input ( c ), but it does not provide any updated belief about ( z ). Instead, it's used to assess how well a particular ( z ) and ( c ) explain the data ( y ).

2. What is the Posterior ( p(z | y, c) )?

The posterior ( p(z | y, c) ) is the probability distribution over the latent variable ( z ), given the observed data ( y ) and the known input ( c ).

• Posterior: This is the updated belief about ( z ) after observing both the data ( y ) and the input ( c ). It incorporates prior knowledge (through ( p(z | c) ), the prior) and the evidence (through the likelihood ( p(y | z, c) )) using Bayes' theorem:

[ p(z | y, c) = ]

The posterior gives us a revised or updated understanding of the latent variable ( z ) after considering the observed data ( y ).

3. The Role of the Likelihood and Posterior in Bayesian Inference

• The likelihood ( p(y | z, c) ) tells us how likely the observed data ( y ) is, given specific values of the latent variable ( z ) and the known input ( c ).
• The posterior ( p(z | y, c) ) is the result of updating our prior belief about ( z ) using the observed data ( y ) (via the likelihood) and the prior ( p(z | c) ).

Why the Difference Matters:

• The likelihood ( p(y | z, c) ) is a function of the observed data ( y ) given the latent variable ( z ). It is used to compute the posterior but does not describe the distribution of ( z ).
• The posterior ( p(z | y, c) ) is the final distribution that tells us how our belief about ( z ) has changed after observing ( y ).

In short, ( p(y | z, c) ) is the likelihood, not the posterior, because it tells us how likely the data is, given ( z ) and ( c ). The posterior ( p(z | y, c) ) updates our knowledge about ( z ) based on both the prior ( p(z | c) ) and the likelihood ( p(y | z, c) ).