# Is this formula related to the energy function / boltzmann distribution?

I am reading research done in this thesis. On page 20-21 (section 2.1) of the thesis, it describes the following stochastic model:

... let us focus for the rest of this chapter on time-varying signals ... In this context, an observed object is simply a signal in a time window of a certain length. A latent source object may be thought of as a signal corresponding to a prototypical type of event. If the same type of event were to happen many times, we suppose that the resulting observed signals are noisy versions of the latent source signal corresponding to that type of event.

We say an observation $$s$$ from an infinite stream $$s_{\infty}$$, is generated by a latent source $$q$$ if $$s$$ is a noisy version of $$q$$. Accordingly, we propose the following stochastic model relating a latent source $$q$$ and an observation $$s$$: $$P(s \text{ generated by } q) \propto \text{exp}(-\gamma d(s,q))$$

where $$d(s, q)$$ is the distance between $$s$$ and $$q$$ and $$-\gamma$$ is a scaling parameter.

I had never seen this formula before, and after looking around, I think it may be related to a potential function, whose representation is the Boltzmann distribution. There is a picture on page 23 of the thesis depicting what seems like different potential values.

Am I correct? Any other insights/references are welcome.

The distribution that you have, when using the squared distance $$d (s,q) = \sum_{i=1}^{N_{obs}} (s_i-q_i)^2$$ looks more like a multivariate Gaussian distribution.
When you substitute that term then the equation becomes: $$P(s \text{ generated by } q) \propto \prod_{i=1}^{N_{obs}} \text{exp}(-\gamma (s_i-q_i)^2)$$