# Thoughts from Friday 3/11

## Overarching research questions

### What is a good basis?

BUT - this is dependent on where the data is

EX1. consider any basis that is infinitely differentiable in a region, then if we consider points close enough, we'll obtain piecewise linear and will not obtain good uncertainties nor fit (we think)

- if this is true, this tells us that we need the in-distribution and out-of-distribution areas of the data in order to meaningfully ask this question

This leads us to...

### What is a good basis given regions of in-distribution (ID) and out-of-distribution (OOD) data?

But... given any good basis, can we always choose an adversarial dataset that gives us poor uncertainty?

EX2. If we are given a "good basis", but then we choose $y = B_1(x)$ (exactly the first basis function) as our data, won't we get poor uncertainty? because maybe the weights will just be on the first basis function.

- this may be a specific example of the more general phrase:
- for all bases, exists a dataset such that the basis is "bad" for that dataset
- if this is true, we need more than just ID and OOD areas - we need more properties about the data

This leads us to...

### What is a good basis given a particular dataset (or some general characteristics about the data) and ID and OOD regions (OOD -> where we are measuring the uncertainty)?

- diversity??
- RFF gives you a basis which is good
- want to test with different datasets like exp, xsinx, cubic, sinx

Rephrasing the question:

### Is there a metric to distinguish good/bases given a dataset?

#### What's wrong with just getting data, fitting the NLM, then reporting the uncertainty?

- this is tangential to our problem, which is analyzing a good bases