What does "regularity condition" mean

Question

While reading a blog post about score matching, I came across the term 'regularity condition'. When I searched for it on Google, I found an answer:

The regularity condition refers to a restriction imposed on the likelihood function in order to ensure that the order of expectation operation and differentiation can be interchanged.

However, I don't find this definition to make sense in the context of the blog post and the topic of score matching. So, what does 'regularity condition' actually mean?

Edit : I added direct link to the quote. The author mentioned that term in context of explaining why we can drop two first term in the equation below :

\begin{align*} &\underbrace{-\lim_{b \to \infty} \nabla_x \log p_m(b; \theta) p_d(b)}_{0} + \underbrace{\lim_{a \to -\infty} \nabla_x \log p_m(a; \theta) p_d(a)}_{0} \\ &+ \int_{-\infty}^{\infty} \nabla_x^2 p_m(x; \theta) p_d(x) \, dx. \end{align*}

Hyvärinen et al. make the assumption (this is a regularity condition in their Theorem 1) that for any $\theta$,

\begin{equation} p_d(x) \nabla_x \log p_m(x; \theta) \to 0 \quad \text{when} \quad \|x\|_2 \to \infty, \end{equation}

which allows us to drop the first two terms.

Answer 1

"Regularity condition" is a general term for conditions that make functions or variables well-behaved enough so that proofs work. One example is conditions that allow differentiation and integration to be interchanged, but another is conditions on existence or finite moments of derivatives of a function, such as the smoothness conditions on the log-likelihood for Cramér's proof that the MLE is asymptotically Normal. There's no precise rule about what counts as a regularity condition; for example, a condition on the rate of decay with distance in a spatial process might be called a regularity condition or it might not. I would be more likely to call something a 'regularity condition' if it wasn't the main aspect of the proof that I was interested in, and there's an implication that "reasonable" examples will satisfy the regularity condition. Saying "mild regularity condition" makes that implication explicit.

In your example, the term "regularity condition" is claiming (a) if you make this assumption the proof is easier because those terms go away and at least implicitly (b) that most of the $p_d$ or $p_m$ a reasonable person would care about do satisfy that condition.

The term "regular", like "normal" is used a lot in mathematics to indicate some sort of good behaviour, with the implication that it's not unreasonable. In statistics, in addition to the informal use above we have regular estimator for an estimator having a limit that is locally a location shift). There are also unrelated uses of regular in measure theory (regular measure), topology (regular space), set theory (regular cardinal),group theory (regular representation), category theory (regular category), and probably lots of others.