标签：#latentvariables

A probabilistic model can describe more than the data you observe. It can also include hidden variables that capture structure you cannot observe directly. But defining that model is only the beginning. Once an observation x is available, the practical question changes: Given this x , what does the model imply about the hidden variable z ? That is the central problem of Posterior Inference . The notation is compact, but the computation is not always easy. High-dimensional latent spaces, complex posterior distributions, and interactions among hidden variables can make both the posterior itself and expectations under that posterior difficult to compute. Start with the Joint Distribution Suppose a probabilistic model contains an observed variable x and a hidden or latent variable z . The model does not treat them as unrelated quantities. Instead, it represents their probabilistic relationship through a Joint Distribution : p ( z , x ) This joint distribution describes how the observed data and the hidden variable fit together inside a single probability structure. Once x is observed, however, the question becomes conditional. We are no longer asking only how x and z relate in general. We want to know how the possible values of z are distributed given the particular observation x . That conditional distribution is the posterior. Posterior Distribution: Conditioning on Observed Data The Posterior Distribution is p ( z ∣ x ) = p ( x ) p ( z , x ) ​ The numerator p ( z , x ) contains the probabilistic relationship between the latent variable and the observation. The denominator p ( x ) normalizes those values so that the result becomes a conditional probability distribution over z . The distinction is important: The joint distribution p ( z , x ) describes the probability structure of the model. The posterior distribution p ( z ∣ x ) tells us what that structure implies about z after x has been observed. In that sense, the posterior connects the model with actual data. Pos