# [[latent variable]] a [[statistical model]]: we model the [[covariate|observed data]] from a ([[parameter]]ized) [[conditional density]] $\boldsymbol{x} \sim f[\theta_{\text{X}}, \boldsymbol{z}]$ where $\boldsymbol{z} \sim \pi[\theta_{\text{Z}}]$ are the unobserved [[dimensionality reduction|latent variable]]s that generate the observation. *perceptual aliasing* issue: case when two distinct $\boldsymbol{z}_{0}, \boldsymbol{z}_{1}$ generate the same observed data $\boldsymbol{x}$ (ie $f$ is not [[injective]] in deterministic case) # cf [[supervised]] learning [[latent variable]]s are *unobserved*. instead, to learn the relationship between *observed* [[covariate]] variables and *observed* [[response]] variables, see [[supervised]] learning; [[signal noise decomposition]] and [[additive noise]] for an example assumption of the [[probability density function|likelihood]] # details [[Bayes rule]] gives the [[posterior]] (for a single data point $\boldsymbol{x}$) $ \tilde{\pi}[\theta, \boldsymbol{x}](\boldsymbol{z}) = \frac{f[\theta_{\text{X}}, \boldsymbol{z}](\boldsymbol{x}) \, \pi[\theta_{\text{Z}}](\boldsymbol{z})}{\Pr(\boldsymbol{x} \mid \theta)} $ ^posterior note this nice relationship between 1. [[covariate|observed data]] [[probability density function|log likelihood]], 2. [[Kullback-Leibler divergence|kld]] from the [[variational inference|variational approximation]] to the [[posterior]], 3. the [[evidence lower bound|elbo]]: ![[evidence lower bound#^elbo-decomposition]] # observation in context of [[stochastic process]]: often there is some "underlying" [[stochastic process]] $s_{t}$ but we only observe some function $o_{t} = O(s_{t}, \omega_{t+1})$ where $\omega_{t+1}$ is [[exogeneous]] information (typically [[noise]]). such a process is called a [[hidden Markov model]] or a [[partially observed Markov decision process]] in the [[sequential decision making]] case. otherwise (if we observe the [[state]]) we just have a [[Markov chain]] / [[Markov decision process]].