How linear regression works intuitively and how it leads to gradient descent | Hacker News Reader

How linear regression works intuitively and how it leads to gradient descent

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334 pointslucasfcosta101 comments

One interesting property of least squares regression is that the predictions are the conditional expectation (mean) of the target variable given the right-hand-side variables. So in the OP example, we're predicting the average price of houses of a given size.

The notion of predicting the mean can be extended to other properties of the conditional distribution of the target variable, such as the median or other quantiles [0]. This comes with interesting implications, such as the well-known properties of the median being more robust to outliers than the mean. In fact, the absolute loss function mentioned in the article can be shown to give a conditional median prediction (using the mid-point in case of non-uniqueness). So in the OP example, if the data set is known to contain outliers like properties that have extremely high or low value due to idiosyncratic reasons (e.g. former celebrity homes or contaminated land) then the absolute loss could be a wiser choice than least squares (of course, there are other ways to deal with this as well).

Worth mentioning here I think because the OP seems to be holding a particular grudge against the absolute loss function. It's not perfect, but it has its virtues and some advantages over least squares. It's a trade-off, like so many things.

[0] https://en.wikipedia.org/wiki/Quantile_regression

Yeah. Squared error is optimal when the noise is Gaussian because it estimates the conditional mean; absolute error is optimal under Laplace noise because it estimates the conditional median. If your housing data have a few eight-figure outliers, the heavy tails break the Gaussian assumption, so a full quantile regression for, say, the 90th percentile—will predict prices more robustly than plain least squares.

True. But it's worth mentioning that normality is only required for asymptotic inference. A lot of things that make least squares stand out, like being a conditional mean forecast, or that it's the best linear unbiased estimator, hold true regardless of the error distribution.

My impression is that many tend to overestimate the importance of normality. In practice, I'd worry more about other things. The example in the OP, eg, if it were an actual analysis, would raise concerns about omitted variables. Clearly, house prices depend on more factors than size, eg location. Non-normality here could be just an artifact of an underspecified model.

How does an upcoming college student, or worse an already graduate, learn statistics like this, with depth of understanding of the meaning of the math, vs just plug an chugging cookbook formulas and "proving" theorems mechanically without the deep semantics?

Statistical Rethinking is quite good in explaining this stuff. https://xcelab.net/rm/

Basically all of the Andrew Gelman books are also good.

Data Analysis... https://sites.stat.columbia.edu/gelman/arm/ Regression and Other Stories: https://avehtari.github.io/ROS-Examples/

Wasserman's All of Statistics is a really good introduction to mathematical statistics (the Gelman stuff above are more practically and analytically focused).

But yeah, it would probably be easier to find a good statistics course at a local university and try to audit it or do it at night.

Dont take the “for engineers” version.

> and "proving" theorems mechanically

I think you’ve have a bad experience because writing a proof is explaining deep understanding.

> I think you’ve have a bad experience because writing a proof is explaining deep understanding.

I think your wording is the key—coming up with a proof is creating deep understanding, but writing a proof very much need not be explaining or creating deep understanding. Writing a proof can be done mechanically, by both instructor and student, and, if done so, neither demonstrates nor creates understanding.

(Also, in statistics more than in almost any other mathematically based subject, while the rigorous mathematical foundations are important, a complete theoretical understanding of those foundations need not shed any light on the actual practice of statistics.)

> need not shed any light on the actual practice of statistics.

That’s not what this comment asked for.

You're right. Coming up with a proof is a creative process. Each major proof in mathematics is so unique, that it usually gets named after its inventor. So we have Euclid's proof that there are infinitely many primes, Euler's proof that e is irrational, and Wiles' proof of Fermat's last theorem.

I'd say reading about statistics and being curious is a great start :)

Quantile regression is great, especially when you need more than just the average. A quantile model for, say, the 10th and 90th percentiles of something are really useful for decision-making. There is a great R package called qgam that lets you fit very powerful nonlinear quantile models -- one of R's "killer apps" that keeps me from using Python full-time.

This is very light and approachable but stops short of building the statistical intuition you want here. They fixate on the smoothness of squared errors without connecting that to the gaussian noise model and establishing how that relates to the predictive power against natural sorts of data.

Any resource/link you know of that further develops your point?

CMU lecture notes [0] I think approach it in an intuitive way, starting from the Gaussian noise linear model, deriving log-likelihood, and presenting the analytic approach. Misses the bridge to gradient methods though.

For gradients, Stanford CS229 [1] jumps right into it.

[0] https://www.stat.cmu.edu/~cshalizi/mreg/15/lectures/06/lectu...

[1] https://cs229.stanford.edu/lectures-spring2022/main_notes.pd...

Thanks! will have a look..

Yes, and it seems like it could’ve been written in-part by an LLM. But, the LLM could take your criticism, improve upon the original, and iterate that way until you feel that it has produced something close to an optimal textbook. The one thing missing is soul. I noticeably don’t feel like there was anyone behind this writing.

Ah, we’re resorting to ad machinum today. :)

It isn't too hard to find resources on this for anyone genuinely looking to get a deeper understanding of a topic. I think a blog post (likely written for SEO purposes, which is in no way a knock against the content) is probably the wrong place that kind of enlightenment, but I also think there are limits to the level of detail you can reasonable expect from a high level blog post.

And for introductory content there's always that risk if you provide to much information you overwhelm the reader, make them feel like maybe this is too hard for them.

Personally I find the process of building a model is a great way of learning all this.

I think a course is probably helpful, but the problem with things like data camp is they are overly repetitive and they don't do a great job of helping you look up earlier content unless you want to scroll through a bunch of videos, where the formula goes on screen for 5 seconds.

Would definitely just recommend getting a book for that stuff, I found "All of statistics" good, I just wouldn't recommend trying to read it from cover to cover, but I have found it good as a manual where I could just look up the bits I needed when I needed it. Tho the book may be a bit intimidating if you're unfamiliar with integration and derivatives (as they often express the PDF/CDF of random variables in those terms).

>I think a blog post... is probably the wrong place that kind of enlightenment

There's this site full of cool knowledgeable people called Hacker News which usually curates good articles with deep intuition about stuff like that. I haven't been there in years, though.

I really recommend this explorable explanation: https://setosa.io/ev/ordinary-least-squares-regression/

And for actual gradient descent code, here is an older example of mine in PyTorch: https://github.com/stared/thinking-in-tensors-writing-in-pyt...

Google search is evil by not giving me those resources.

This is an all-time great blog post for this line alone: "That's why we have statistics: to make us unsure about things."

The interactive visualizations are a great bonus though!

Kagi FTW?

That was my initial thought, too. But I didn't know what the original Google search consisted of and the site didn't show up in a couple Kagi searches I tried. (Aside from the obvious titular one, of course.)

Yeah - I wanted to post it here, but after searching for "linear regression explorable explanation" I got some other random links. Thankfully, I saved the PyTorch materials + https://pinboard.in/u:pmigdal/t:explorable-explanation.

Google does however provide this very nice course that explains these things in more detail: https://developers.google.com/machine-learning/crash-course

Some important context missing from this post (IMO) is that the data set presented is probably not a very good fit for linear regression, or really most classical models: You can see that there's way more variance at one end of the dataset. So even if we find the best model for the data that looks great in our gradient-descent-like visualization, it might not have that much predictive power. One common trick to deal with data sets like this is to map the data to another space where the distribution is more even and then build a model in that space. Then you can make predictions for the original data set by taking the inverse mapping on the outputs of the model.

What you’re describing is the technique known as the “kernel trick”, correct?

No, the kernel trick is something else: basically a nonlinear basis representation of the model. For example, fitting a polynomial model, or using splines, would effectively be using the "kernel trick" (though only ML people use that term, not statisticians, and usually they talk about it in the context of SVMs but it's fine for linear regression too). Transforming the data is just transforming the Y-outcome, most commonly with log(y) for things that tend to be distributed with a right-skew: house prices being a classic example, along with things like income, various blood biomarkers, or really anything that cannot go below zero but can (in principle) be arbitrarily large.

In a few rare cases I have found situations where sqrt(y) or 1/y is a clever and useful transform but they're very situational, often occurring when there's some physical law behind the data generation process with that sort of mathematical form.

To be fair, the "trick" part of the kernel trick involves implicitly transforming the data into a higher dimensional space and then fitting a linear function in that space. Ie, you're transforming the inputs so that a linear function from inputs to outputs fits better than if you didn't do the transform.

The "trick" allows you to fit a linear function in that higher dimensional space without any potentially costly explicit computation in the higher dimensional space based on the observation that the optimal solution's parameters can be represented as a sum of the higher dimensional representations of points in the training set.

No actually I think you’re mistaken. Representing the model via a nonlinear transformation where a linear model more closely captures what’s going on is precisely what the kernel trick does, although the situation being described is more broad than the kernel trick, things like the power transform also fit the bill.

The kernel trick is a technique used in data classification that involves mapping the points into a higher dimensional space and then finding a linear separation in that higher dimension.

It's not about finding a line of best fit or making the dataset appear linear, it's about being able to split a dataset into two classes using a linear function.

Sure, it’s not about finding a line of best fit, but the principle is the same: a transformed space where linear things work better is used.

Just keep in mind, the kernel trick is a way to transform a data set so that "linear things work better"... although that's very vague I mean sure it's passable but it's also different from what was originally posted... the kernel trick doesn't transform your data into a space where that data becomes linear. It transforms your data into a space where it can be separated by a line/plane. The data is almost always non-linear in that transformed space but it's transformed in a way that a plane can cleanly separate that data.

Given that the kernel trick is pretty specific jargon used mostly in a specific circumstance, it's in your interest to use that term in that specific context. If you're interested in the more general term of making things work with respect to some function, which can be linear or Gaussian or some other form the term is "feature transformation".

In my work, I hardly ever use linear regression, but do use multiple linear regression. Multiple linear regression allows multiple linear predictors, where the method parses shared and independent variances associated with each predictor. These discussions on linear regression hardly ever touches on the very useful multiple linear regression method. In the case of bad variance inflation in models with multi-collinear predictors, robust regression techniques are advised like ridge, LASSO, or elastic net regression.

In relation to gradient descent, I do not know enough if multiple regression is at all relevant, or why not.

And yeah, for non-normal error distributions, we should be looking at generalized linear models, which allows one to specify other distributions that might better fit the data.

Non-constant variance does not actually bias the coefficients of a linear regression model -- thus, its predictions will be just fine. What it does is underestimate the standard errors; your p-values will typically be too small. Sometimes a log-transform or similar can help, but otherwise you can use weighted least-squares.

This kind of problem is actually a good intro to iterative refitting methods for regression models: How do you know what the weights should be? Well, you fit the initial model with no weights, get its residuals, use those to fit another model, rinse and repeat until convergence. A good learning experience and easy to hand-code.

The main practical reason why square error is minimized in ordinary linear regression is that it has an analytical solution. Makes it a bit weird example for gradient descent.

There are plenty of error formulations that give a smooth loss function, and many even a convex one, but most don't have analytical solutions so they are solved via numerical optimization like GD.

The main message is IMHO correct though: square error (and its implicit gaussian noise assumption) is all too often used just per convenience and tradition.

OLS is a straightforward way to introduce GD, and although an analytic solution exists it becomes memory and IO bound at sufficient scale, so GD is still a practical option.

Computationally OLS is taking the pseudoinverse of the system matrix, which for dense systems has a complexity of O(samples * parameters^2). For some GD implementations the complexity of a single step is probably O(samples * parameters), so there could be a asymptotic benefit, but it's hard to imagine a case where the benefit is even realized, let alone makes a practical difference.

And in any case nobody uses GD for regressions for statistical analysis purposes. In practice Newton-Raphson or other more complicated schemes (with a lot higher computation, memory and IO demands) with a lot nicer convergence properties are used.

Mini batch and streaming GD make the benefits obvious and trivial. Closed form OLS is unbeatable so long as samples * params^2 is comfortably sitting in memory. You often lose that as soon as your p approaches 10^5, which is common these days. Soon as you need distributed, streaming, or your data is too tall and or too wide then first order methods are the point of call.

With batching it becomes SGD. If you're OK with approximations, you have e.g. randomized, reduced rank and streaming SVDs. And these tend have a lot nicer approximation and convergence properties than SGD.

What are the common cases for 10^5 parameter OLS? Perhaps something like weather models could include such computations?

This isn't true. In practice people don't use the analytical solution for efficient linear regression, they use stochastic methods.

Square error is used because it is the maximum likelihood estimator under the assumption that observation noise is normally distributed, not because it is analytical.

AFAIK using the analytic solution for linear regression (via lm in R, statsmodels in python or any other classical statistical package) is still the norm in traditional disciplines such as social (economics, psychology, sociology) and physical (bio/chemistry) sciences.

I think that as a field, Machine Learning is the exception rather than the norm, where people people start off or proceed rapidly to non-linear models, huge datasets and (stochastic) gradient based solvers.

Gaussianity of errors is more of a post-hoc justification (which is often not even tested) for fitting with OLS.

If by stochastic methods you mean something like MCMC, they are increasing in popularity, but still used a lot less than analytical or numerical methods. And almost exclusively only for more complicated models than basic linear regression. Sampling methods have major downsides, and approximation methods like ADVI are becoming more popular. Though sampling vs approximations is a bit off topic, as neither usually have closed form solutions.

Even the most popular more complicted models like multilevel (linear) regression make use of the mathematical convenience of the square error, even though the solutions aren't fully analytical.

Square error indeed gives estimates for normally distributed noise, but as I said, this assumption is quite often implicit, and not even really well understood by many practitioners.

Analytical solutions for squared errors have a long history for more or less all fields using regression and related models, and there's a lot of inertia for them. E.g. ANOVA is still the default method (although being replaced by multilevel regression) for many fields. This history is mainly due to the analytical convenience as they were computed on paper. That doesn't mean the normality assumption is not often justifiable. And when not directly, the traditional solution is to transform the variables to get (approximately) normally distributed ones for analytical solutions.

It’s not because of analytical convenience, it’s because of the central limit theorem.

Not everything is a linear combination of large number of (IID) samples, and thus not everything is gaussian distributed.

You’re implying that many things are though.

Yes, and I was explicit about it in another comment in this post.

Ok, so we all agree that in most cases the reason to minimize square error is that it’s the appropriate thing to minimize - not that it has an analytical solution, convenience or tradition.

...because stochastic methods are implicit regularizers, leading to solutions that generalize better. Let's spell it out for those that don't know.

https://www.inference.vc/notes-on-the-origin-of-implicit-reg...

OLS is a convex optimization problem, so this doesn't really apply. And for statistical analysis you really don't want to add poorly understood artificial noise to the parameter estimates anyway.

In general you do, because the unbiased estimates have higher generalization error. You are already dealing with sampling noise. I am not an expert in optimization, and what "poorly understood" means to you, but I know there is quite some research on the properties of SGD noise; e.g., https://francisbach.com/rethinking-sgd-noise/

Dissecting the Effects of SGD Noise in Distinct Regimes of Deep Learning https://arxiv.org/abs/2301.13703

I’ve always felt that ML introductions completely butcher OLS. When I was taught it in stats we had to consider the Gauss-Markov conditions and interpret the coefficients, we would study the residuals. ML introductions just focus getting good predictions.

IMO that's the fundamental difference between statistics and ML. The culture of stats is about fitting a model and interpreting the fit, while the culture of ML is to treat the model as a black box.

That's one of the reasons that multicollinearity is seen as a big deal by statisticians, but ML practitioners couldn't give a hoot.

You are describing the difference between academic mathematician statisticians and "applied/engineering/actuarial/business" people who use statistics. The "black box" culture goes back to before ML and before both computing Machines M and statistical Learning (iterative models)

I suspect that the "black box" philosophy for statistics/ML is actually bad if you don't have a quick way of verifying the predictions. For instance, using PCA as a "black box" is perfectly fine if you're using it to de-noise readings from a camera or other instrument, because a human being can quickly tell if the de-noising is working correctly or not. But if you're using PCA to make novel discoveries, where you don't have an independent way of checking those discoveries, then it might be outright essential to have a deep definition-theorem-proof style understanding of PCA. What do people think of this hunch?

The point about PCA applies to population genetics and psychometrics (IQ). Some conclusions have been derived using PCA that appear to be supported by little else, and these have come under question.

You make a good point, though the difference between ML and statistics isn't just about interpreting and validating the model. It's about the "novel discoveries" part aka Doing Science.

Statistical modeling is done primarily in service of scientific discovery--for the purpose of making an inference (population estimate from a sample) or a comparison to test a hypothesis derived from a theoretical causal model of a real-world process before viewing data. The parameters of a model are interpreted because they represent an estimate of a treatment effect of some intervention.

Methods like PCA can be part of that modeling process either way, but analyzing and fitting models to data to mine it for patterns without an a priori hypothesis is not science.

Only perfect multicollinearity (correlation of 1.0 or -1.0) is a problem at the linear algebra level when fitting a statistical model.

But theoretically speaking, in a scientific context, why would you want to fit an explanatory model that includes multiple highly (but not perfectly) correlated independent variables?

It shouldn't be an accident. Usually it's because you've intentionally taken multiple proxy measurements of the same theoretical latent variable and you want to reduce measurement error. So that becomes a part of your measurement and modeling strategy.

I think this distinction is not sharp. You do hear ML practitioners talk about interpretability a lot.

That is incorrect. Least squares follows directly from the central limit theorem.

Central limit theorem tells in practice that gaussian distributions is can be expected to be quite common. And it makes the gaussian distribution a good first guess. Least squares gives the ML estimate for gaussian residuals. I don't find this very direct, and there being a rationale doesn't mean that rationale is what in reality drives the usage.

I mention the relation to the gaussian distribution. Which part of the comment is incorrect?

This part is incorrect: “ The main practical reason why square error is minimized in ordinary linear regression is that it has an analytical solution”

OLS is popular because it gives correct answers as a result of the CLT

And it has an analytical solution, which was important before computing (and still makes it quicker today).

In other words, as economists say, because OLS is provably the BLUE (Best Linear Unbiased Estimator) aka the Gauss-Markov Theorem.

> When using least squares, a zero derivative always marks a minimum. But that's not true in general ... To tell the difference between a minimum and a maximum, you'd need to look at the second derivative.

It's interesting to continue the analysis into higher dimensions, which have interesting stationary points that require looking at the matrix properties of a specific type of second order derivative (the Hessian) https://en.wikipedia.org/wiki/Saddle_point

In general it's super powerful to convert data problems like linear regression into geometric considerations.

Mathematical ignoramus writing here, but I have a long-term project to correct my ignorance of statistics so this seems a good place to start.

He isn't talking about how to calculate the linear regression, correct? He's talking about why using squared distances between data points and our line is a preferred technique over using absolute distances. Also, he doesn't explain why absolute distances produce multiple results I think? These aren't criticisms, I am just trying to make sure I understand.

ISTM that you have no idea how good your regression formula (y = ax + c) is without further info. You may have random data all over the place, and yet you will still come out with one linear regression to rule them all. His house price example is a good example of this: square footage is, obviously, only one of many factors that influence price -- and also the most easily quantified factor by far. Wouldn't a standard deviation be essential info to include?

Also, couldn't the fact that squared distance gives us only one result actually be a negative, since it can so easily oversimplify and therefore cut out a whole chunk of meaningful information?

I intuitively think about linear regression as attaching a spring between every point and your regression line (and constraining the spring to be vertical). When the line settles, that's your regression! Also gives a physical intuition about what happens to the line when you add a point. Adding a point at the very end will "tilt" the line, while adding a point towards the middle of your distribution will shift it up or down.

A while ago I think I even proved to myself that this hypothetical mechanical system is mathematically equivalent to doing a linear regression, since the system naturally tries to minimize the potential energy.

Perfect analogy! The cool part is that your model also gives good intuition about the gradient descent part. The springs' forces are the gradients, and the act of the line "snapping" into place is the gradient descent process.

Technically, physical springs will also have momentum and overshoot/oscillate. But even this is something that is used in practice, gradient descent with momentumg.

I don't have anything useful to say, but, how the hell is that a "12 min read"?

I always find those counters to greatly overestimate reading speed, but for a technical article like this it's outright insulting, to be honest.

It's the common trap of trying to teach, and why teaching is so much more difficult than it appears.

When you intimately understand a topic, you have an intuition that naturally paves over gaps and bumps. This is excellent for getting work done, but terrible for teaching. Your road from start to finish is 12 minutes, and without that knack for teaching, you are unable to see what that road looks like to a beginner.

I built a small static web app [0] (with svelte and tensorflow js) that shows gradient descent. It has two kind of problems: wave (the default) and linear. In the first case, the algorithm learns y = ax + b ; in the second, y = cos(ax + b). The training data is generated from these functions with some noise.

I spent some time making it work with interpolation so that the transitions are smooth.

Then I expanded to another version, including a small neural network (nn) [1].

And finally, for the two functions that have a 2d parameter space, I included a viz of the loss [2]. You can click on the 2d space and get a new initial point for the descent, and see the trajectory.

Never really finished it, though I wrote a blog post about it [3]

[0] https://gradfront.pages.dev/

[1] https://f36dfeb7.gradfront.pages.dev/

[2] https://deploy-preview-1--gradient-descent.netlify.app/

[3] https://blog.horaceg.xyz/posts/need-for-speed/

> It has two kind of problems: wave (the default) and linear. In the first case, the algorithm learns y = ax + b ; in the second, y = cos(ax + b).

Are "first" and "second" switched here?

All thats wrong with the modern world

https://www.ibm.com/think/topics/linear-regression

A proven way to scientifically and reliably predict the future

Business and organizational leaders can make better decisions by using linear regression techniques. Organizations collect masses of data, and linear regression helps them use that data to better manage reality, instead of relying on experience and intuition. You can take large amounts of raw data and transform it into actionable information.

You can also use linear regression to provide better insights by uncovering patterns and relationships that your business colleagues might have previously seen and thought they already understood.

For example, performing an analysis of sales and purchase data can help you uncover specific purchasing patterns on particular days or at certain times. Insights gathered from regression analysis can help business leaders anticipate times when their company’s products will be in high demand.

While I get your point, it doesn't carry too much weight, because you can (and we often read this) claim the opposite:

Linear regression, for all its faults, forces you to be very selective about parameters that you believe to be meaningful, and offers trivial tools to validate the fit (i.e. even residuals, or posterior predictive simulations if you want to be fancy).

ML and beyond, on the other hand, throws you in a whirl of hyperparameters that you no longer understand and which traps even clever people in overfitting that they don't understand.

Obligatory xkcd: https://xkcd.com/1838/

So a better critique, in my view, would be something that the JW Tukey wrote in his famous 1962 paper: (paraphrasing because I'm lazy):

"better to have an approximate answer to a precise question rather than an answer to an approximate question, which can always be made arbitrarily precise".

So our problem is not the tools, it's that we fool ourselves by applying the tools to the wrong problems because they are easier.

That particular xkcd was funny until the LLMs came around

Aren't LLMs also a pile of linear algebra?

And a pinch of ReLU

That's the point, yes. "Piling up more and more data then stirring it until it works" stopped being a joke and turned out to be a practical approach.

This can be seen as another occurence of the "bitter lesson": http://www.incompleteideas.net/IncIdeas/BitterLesson.html

Uh, it's funny because it works. It came out at a point where that approach was already being used in plenty of applications.

Thanks for the link to The Bitter Lesson.

I indeed find the lesson that it describes unbearably bitter. Searching and learning, as used by the article, may discover patterns and results (due to infinite scaling of computation) that we, humans, are physically uncapable of discovering -- however, all those learnings will have no meaning, they will not expose any causality. This is what I find unbearable, as it implies that the real world must ultimately remain impervious to human cognizance; it implies that our meaning- and causality-based human reasoning ultimately falls short to model the world, while general, computation-only methods (given ever-growing computing power) at least "converges" to a faithful (but meaningless) description of the world.

See examples like protein folding, medicine research, AI-assisted diagnosis, self driving cars. We're going to rely on their results, but we'll never know why those results work. We're not going to reject self-driving cars if those cars save lives per same distance driven and/or same time driven; however, we're going to sit in, and drive, those cars blind. To me, that's an unbearable thought, even apart from the possibility that at some point the system might break down, and cause a huge accident inexplicably. An inexplicable misbehavior of the system is of course catastrophic, but to me, even the inexplicable proper behavior of the system is an unsettling thought -- because it is inexplicable.

Edited to add: I think the phrase "how we think we think" is awesome in the essay. We don't even know how our reasoning works, so trying to "machinize" those misconceptions is likely bound to fail.

Arguably, "the way our reasoning works" is probably a normal distribution but with a broad curve (and for some things, possibly a bimodal distribution), so trying to understand "why" is a fool's errand. It's more valuable to understand the input variables and then be able to calculate the likely output behaviors with error bars than to try to reduce the problem to a guaranteed if(this), then(that) equation. I don't particularly care why a person behaves a certain way in many cases, as long as 1) their behavior is generally within an expected range, and 2) doesn't harm themselves or others, and I don't see why I'd care any more about the behavior of an AI-driven system. As with most things, Safety first!

Well I'd say that prompt engineering is still exactly this?

My maxim of statistics is that applied statistics is the art of making decisions under uncertainty, but people treat it like the science of making certainty out of data.

That sums it up exceptionally well.

In the same vein, Karpathy's video series "Neural Networks from zero to hero"[0] touches upon a lot of this and intuitions as well. One of the best introductory series (even if you ignore the neural net part of it) and brushes on gradients, differentiation and what it means intuitively.

[0] https://youtu.be/VMj-3S1tku0?si=jq1cCSn5si17KK1o

See another interactive article explaining linear regression and gradient descent: https://mlu-explain.github.io/linear-regression/

Nice, thanks for sharing! I shared this with my HS calculus teacher :) (My model is that his students should be motivated to get machine learning engineering jobs, so they should be motivated to learn calculus, but who knows.)

Very neat, didn't know about Deming Regression. Hard to imagine a case now where it wouldn't be the more appropriate method.

Speaking of linear regression, can any of you recommend an online course or book that deep dives into fitting linear models?

Most intro to stats courses will do

I did the Stats I -> II -> II pipeline at uni but you should be fitting basic linear models by the end of Stats I

This (housing prices) example seems really familiar. Was it used in Andrew Ng's original Coursera ML class?

Housing price examples in regression are much, much older than Ng’s ML class.

Another way to approach the explanation is understanding the data generating process i.e. the statistical assumptions of the process that generates the data. That can go a long way to understanding _analytically_ if linear regression model is a good fit(or what to change in it to make it work). And — arguably more importantly — also a reason why we frame linear regression as a statistical problem instead of an optimization one(or an analytical OLS) in the first place. I would argue understanding it from a statistical standpoint provides much better intuition to a practitioner.

The reason to look at statistical assumptions, is because we want to make probabilistic/statistical statements about the response variable, like how much is its central tendency and how much it varies as values of X change. The response variable is not easy to measure.

Now, one can easily determine, for example using OLS(or gradient descent), the point estimates for parameters of a line that needs to be fit to two variables X and Y, without using any probability or statistical theory. OLS is, in point of fact, just an analytical result and has nothing to do with theory of statistics or inference. The assumptions of simple linear regression are statistical assumptions which can be right or wrong but if they hold, help us in making inferences, like:

  - Is the response variable varying uniformly over values of another r.v., X(predictors)?

  - Assuming an r.v. Y what model can we make if its expectation is a linear function.

So why do we make statistical assumptions instead of just point estimates? Because all points of measurements can’t be certain and making those assumptions it is one way of quantifying uncertainty.. Indeed, going through history one finds that Regression's use outside experimental data(Galton 1885) was discovered much after least squares(Newton 1795-1809). The fundamental reasons to understand natural variations in data was the original motivation. In Galton's case he wanted to study hereditary traits like wealth over generations as well as others like height, status, intelligence( coincidentally its also what makes the assumptions of linear regression a good tool for studying this: I think it's the idea of Regression to the mean; Very Wealthy or very pool families don't remain so over a families generations, they regress towards the mean. So is the case with Societal Class, Intelligence over generations)

When you follow this arc of reasoning, you come to the following _statistical_ conditions the data must satisfy for linear assumptions to work(ish):

Linear mean function of the response variable conditioned on a value of X

E[Y|X=x] = \beta_0+\beta_1*x

Constant Variance of the response variable conditioned on a value of X

Var[Y|X=x] = \sigma^2 (OR ACTUALLY JUST FINITE ALSO WORKS WELL)

When you frame it as an optimization problem, like by optimizing the squares loss or cross entropy, you have decided that your data generating process(DGP), i.e. Y is:

- A Binomial/Multinomial random variable, which gives you the the cross entropy like loss function.

- Is a Normal random variable, which gives you the squared loss.

This point is where many ML text books skip to directly. Its not wrong to do this, but this is a much more narrow intuition of how regression works!

But there is no reason Y needs to follow those two DGPs (The process could be a poisson or a mean reverting process)! There is no reason to believe prima-facie and apriori that the Y|X is following those assumptions. This also gives motivation for using other kinds of models.

Its why you test weather those statistical assumptions carefully first using a bit of EDA and from it comes some appreciation and understanding of how linear regression actually works.

The amount of em dashes in this make this look very AI written. Which doesn't make it a bad piece but makes me more carefully check every sentence for errors.

I know this is repeated ad nauseam by now, but as an ardent user of em dashes for many years pre-LLM, I think this a bad heuristic.

Co-author and founder of Briefer here.

I used to use em dashes before they were cool. I actually learned about them when I emailed a guy who's a software engineer at Genius and also writes for The New Yorker and The Atlantic.

I asked him for tips on how to write well and he recommended that I read Steven Pinker's "The Sense of Style", which uses em dashes exhaustively, and explains when and why one should use them.

It also pains me that I can't use them anymore or else people will think an AI did the writing.

I also recommend "The Sense of Style"; knowing how to wield punctuation and grammatical structure is critical for clearly and successfully articulating your ideas. I use semicolons, colons, and parentheticals heavily (but en dashes and em dashes are great too).

I also recommend "The Sense of Style"--knowing how to wield punctuation and grammatical structure is critical for clearly and successfully articulating your ideas--and I use semicolons, colons, and parentheticals heavily (but en dashes and em dashes are great too).

I find that dashes are great for conversational style flowing sentence structure, but sometimes they can become too long and tiring to the reader.

As another ardent user I actually think it is a good but unfortunate heuristic.

Previously I rarely saw it used in my English-as-second-language peer group, even by otherwise decent writers. Now I see it everywhere in personal/professional updates in my feed by. The simpler assumption is that people over-rely on LLMs for crafting these posts, and LLMs disproportionately use em dashes.

This tired take is in every thread now. The sort of behavior better served by a Reddit bot, and just as annoying.

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