There are entire statistics textbooks devoted to multilevel linear models, you can get really far with these.
Shrinking through information sharing is really important to avoid overly optimistic predictions in the case of little data.
[1] https://cran.r-project.org/web/packages/ashr/index.html [2] https://github.com/mengyin/vashr
R's mgcv package (which does all of the above) is probably the single reason I'm still using R as my primary stats language.
There are absolutely decisions that need to get made, and do get made, that are not linear. Step functions are a great example. "We need to decide if we are going to accept this acquisition offer" is an example of a decision with step function utility. You can try to "linearize" it and then apply a threshold -- "let's agree on a model for the value at which we would accept an acquisition offer" -- but in many ways that obscures that the utility function can be arbitrarily non-linear.
That 0/1 input variable could also have arbitrary interactions with other variables, which would also amount to “step function “ input effects.
See for example the autism/age setup down thread.
If all the numbers that appear in your constraints are rational (p/q with finite p and q), then any solution is also a rational number (with finite nominator and finite denominator).
(Well, any finite solution. Your solution could also be unbounded, then you might have infinities in there.)
A computer can represent finite rational numbers just fine. See eg https://docs.python.org/3/library/fractions.html or https://hackage.haskell.org/package/base-4.20.0.1/docs/Data-... for some libraries.
Though in most cases, people just use floating point numbers in practice, but that's of no philosophical concern.
But the parent comment is not talking about constrained optimization, just gradient following.
In the context of this post, that’s just “which of these N discrete variables, if moved from 0 to 1, will increase the quantity of interest according to the linear model?” “Which will decrease it?”
The question is not, “if I can only set M of these N variables to 1, which should I choose?”
That’s a good question, and it leads to problems in NP, but that’s not what the comment was referring to.
Yes, you are right in that abstract setting.
If you always have the full hypercube of available, the problem is as easy as you describe. But if there are constraints between the variables, it gets hairier.
What I think is the important part, is that it is better to ask decision makers for decisions on setting a continuous parameter, than to make binary yes/no or go/no-go decisions. When it's a decision by committee, I can see why that is.
For simplicity, I’m going to assume each variable in the model is independent of every other variable.
We can interpret the coefficients in linear models. This relationship holds for the model for the range of values it is based on. This relationship is the same for the whole range of the model. (We can’t extrapolate outside of what’s been modeled.)
y = c1x1 + c2x2 +…+ cnxn (excuse the poor formatting)
The sign tells you the direction (+ means it will increase the value of y, - means it will decrease the value of y), the value of the coefficient tells you how much the y will change for a given 1-unit change in the x value.
Since this is linear, you get the same change to the output for the relevant increases no matter your starting point.
So, the regression model would say x1, x3, and x5 have positive coefficients and variables x2, x4 have negative coefficients. If you want y to increase, either start doing more of x1, x3, x5 or do less of x2, x4. Depending on what these are and your limited investment budget, for example, you may pick doing x3 if that is the largest positive coefficient.
Again, since this is linear, you can keep on putting resources into the largest coefficient and get the same increase up until your model is no longer valid.
For non-linear models, you can still interpret the coefficients, but the interpretation depends on your starting conditions and where you are on the graph.
There may be asymptotes in your non-linear model, so there is a point of diminishing returns where if you keep putting resources into a variable with a positive coefficient, this will not keep getting you commensurate results.
Sorry I don’t have any actual examples here and I don’t have time to go digging through my old textbooks to look for any.
No, that's not true. Human groups are very able to make discrete decisions. Actually, often they tend to go for discrete decisions, when something continuous (and perhaps linear) would be a lot better.
(Just to be clear: if you force your linear models to make discrete predictions, they are no longer linear in any sense of the word. That's why linear optimisation is a problem that can be solved in polynomial time, and integer linear optimisation is NP complete.
Even convex optimisation, which is no longer linear but still continuous, can be solved in roughly polynomial time.)
Often people demand more decisive decisions, of 'yes'/'no' or concrete action, not shades of grey and fiddling at the margins.
Getting people to even appreciate linear models is already a step forward. Like it or not, your business strategy meetings are already a step ahead of what most people would naturally be inclined to.
And working with these people is so painful.
In my concrete cases I mostly saw that in the direct sense of being able to deploy more mathematics and operations research, eg for netting out (partially) offsetting financial instruments for a bank.
But by introspection you can come up with more example. Eg that's a common selling point for running your servers on AWS instead of building your own hardware.
a) carefully choose the two most important dimensions of concern (as Alan Kay said: the correct point of view is worth 80 Iq points)
b) make them binary: are we happy here or do we need to change?
In a way similar to the pareto ratio, you keep a surprising amount of value in something “so simple it cant be possibly so useful”.
This seems to be getting a lot of attention. I couldn't agree more, we assume linearity all the time because reasoning non-linearly is exceptionally difficult. Yes we can do it sometimes, but it is not the default. Reasoning linearly has its flaws, and we should recognize we are making an imperfect decision, but it is still extremely useful.
Another one I would add that is very important: Human beings, especially in groups, can only reasonably make linear decisions.
That is, when we are in a meeting making decisions for the direction of the company we can only say things like "we need to increase ad spend, while reducing the other costs of acquisition such as discount vouchers". If you want to find the balance between "increasing ad spend" while "decreasing other costs" that's a simple linear model.
Even if you have a great non-linear model, it's not even a matter of "interpretability" so much as "actionability". You can bring the results of a regression analysis to a meeting and very quickly model different strategies with reasonable directional confidence.
I struggled communicating actionable insights upward until I started to really understand regression analysis. After that it became amazingly simple to quickly crack open and understand fairly complex business processes.