# How to understand this probability equation?

$$\mathbf { x } ( t ) = g ( \mathbf { s } ( t ) ; \xi ) + \mathbf { n } ( t )$$, where n(t) denotes the noise or modeling error and ξ the parameters of mapping $$g$$

How to understand the following probability equation?

$$p _ { \mathbf { x } } ( \mathbf { x } ( t ) | \mathbf { s } ( t ) , \xi ) = p _ { \mathrm { n } } ( \mathbf { x } ( t ) - g ( \mathbf { s } ( t ) ; \xi ) )$$, where $$p_n$$ denotes the probability density function of the noise term $$n(t)$$.

It looks reasonable, but how to proof it?

More can be find at section 2.2 Latent variable models https://arxiv.org/pdf/1411.7783.pdf.

## 1 Answer

In words, that equation states that given $$\mathbf{s}(t)$$ and the parameter $$\xi$$, the probability that your random variable/vector (let's call it $$\mathbf{X}$$) takes on the value $$\mathbf{x}(t)$$ is the same as the probability that your noise (let's call that $$\mathbf{N}$$) takes on the value $$\mathbf{x}(t) - g(\mathbf{s}(t);\xi)$$. That is, if we know $$g(\mathbf{S}(t);\xi)$$ (by knowing $$\mathbf{S}$$ and $$\xi$$) then we can find the noise in terms of $$\mathbf{x}$$.

To prove that, we use the definition of the probability mass function (PMF), while noting that $$\mathbf{X}(t) = g(\mathbf{S}(t);\xi) + \mathbf{N}(t)$$ (as a relationship between random variables).

\begin{align} p_{\mathbf{X}}(\mathbf{x}(t) | \mathbf{s}(t), \xi) &= \mathbb{P}(\mathbf{X} = \mathbf{x}(t) | \mathbf{s}(t), \xi) \\ &= \mathbb{P}(\mathbf{X} = g(\mathbf{S};\xi)+\mathbf{N} = \mathbf{x}(t) | \mathbf{S} = \mathbf{s}(t), \xi) \\ &= \mathbb{P}(\mathbf{N} = \mathbf{x}(t) - g(\mathbf{s}(t);\xi) | \mathbf{S} = \mathbf{s}(t), \xi) \\ &= \mathbb{P}(\mathbf{N} = \mathbf{x}(t) - g(\mathbf{s}(t);\xi)) \ (*)\\ &= p_{\mathbf{N}}(\mathbf{x}(t) - g(\mathbf{s}(t);\xi). \end{align}

One thing to note is that going to $$(*)$$ from the equation above assumes that $$\mathbf{S}$$ and $$\xi$$ are independent of $$\mathbf{N}$$.