Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: Has anything in the EA global health sphere changed since the critiques of "randomista development" 1-2 years ago?, published by MaxG on December 2, 2021 on The Effective Altruism Forum.

Just recently I re-discovered posts from 1-2 years ago which discussed critiques of the GiveWell style approach to global health, namely the focus on RCTs.

For example, Ben Kuhn's article suggested that maybe we should look more at the cost-effectiveness of political change. Then there is the wildly popular post on growth and randomista development, which suggested that maybe we should pay more attention to economic growth interventions.

I am sure these articles sparked lots of discussion in the GiveWell and EA global health community. My question is: Has anything changed since then? Has anyone put more resources into researching these topics? How has the thinking of EA global health people developed?

Thanks for listening. To help us out with The Nonlinear Library or to learn more, please visit nonlinear.org.

Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: Omicron Post #3, published by Zvi on December 2, 2021 on LessWrong.

I have split off the Omicron post from the weekly post, which covers other aspects of Covid. I made judgment calls on where to put a few things.

Omicron continues to spread quickly, but the news surrounding Omicron is clearly slowing down, so these updates can hopefully space out a bit more.

Past Omicron posts: #1, #2.

Post on Polymarket’s Covid prediction markets, and potential future markets discussion. Direct link to market on whether Omicron will be >1% of cases by the end of the year, which is currently a moderate favorite.

Biden on Omicron

A few hours after my second Omicron update, Biden went on TV to talk about the new variant.

Here are some highlights:

(1:00) He says that he told us he would always be honest with us, so he’s going to talk about the new Omnicron (which I do admit sounds cooler) variant. Damn that campaign promise, guess we’re going to have to be honest.

(1:00) He says that South Africa’s transparency in telling us about Omnicron is to be applauded, for it allows us to take action, so he took action right away to cut off travel to South Africa. What even are incentives, how do they work, not a metaphor for entire legislative agenda?

(1:30) We can’t stop Omnicron, we can only slow it down, giving us time to take other actions. Therefore get vaccinated. ‘Cause for concern, not cause for panic.’ Because we’re the best, you see, and also science.

(2:30) ‘We have more tools than ever to fight the variant. From vaccines to boosters, to vaccines for children.’ So. vaccines? We have vaccines? ‘And much more.’

(3:00) Did you know ‘deaths from Delta are coming down’? Checks notes.

(3:30) Booster. Vaccinated. Vaccinated. Booster.

(4:30) Mask.

(4:45) This, which I wish I fully believed:

We will now find out what counts as ‘every available tool.’ Also, why aren’t we using ‘every available tool’ with regard to, let’s say, Paxlovid?

(5:30) I will tell the CDC and FDA to use ‘the fastest process available without cutting any corners.’

Wait, if there’s a faster process available that doesn’t cut any corners, then. then.

(5:45) Same thing for ‘modifications to current treatments for those who get ill with the Covid virus.’

(5:50) “Look, I’m removing all roadblocks to keeping the American people safe.”

Paxlovid remains illegal.

(6:05) “If you are vaccinated and worried about this virus, go get your booster.”

Logically this implies that if you have gotten your booster, you should not be worried about this virus. Interesting.

(6:25) Promises a plan for the winter on Thursday that will not use lockdowns, but will use ‘widespread vaccinations, boosters, testing and more.’

(7:00) This statement, the last line of which is pretty rich:

It’s time for the ‘rest of the world’ to ‘step up?’ This is a competition and we’ve set the high score so never mind all the extra doses we could have manufactured, or the ones we’re letting expire, or sit for months in warehouses despite our unwillingness to approve them? It doesn’t matter whether we’ve done our ‘fair share’ or some such nonsense, or who is ‘leading’ it matters what we can do and whether we’re doing it.

He then assures us that no shot given overseas will ever ‘come at the expense of any American’ and he will ensure we are protected first. So get your booster now, to make it easier to send doses overseas?

(9:00) “We’re throwing everything we can at this virus. That’s how we reopen our [various].”

Thing is I think he actually believes that we are throwing everything we can at this virus? And that doing so is ‘how we reopen’ as opposed to deciding to reopen or stay open?

Questions start.

(10:00) Q: Do you think that there will be a future reluctance to report variants due to the reaction this time?

A: No, nor am I going to address the incentive problem at ...

Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: Omicron Post #3, published by Zvi on December 2, 2021 on LessWrong.

I have split off the Omicron post from the weekly post, which covers other aspects of Covid. I made judgment calls on where to put a few things.

Omicron continues to spread quickly, but the news surrounding Omicron is clearly slowing down, so these updates can hopefully space out a bit more.

Past Omicron posts: #1, #2.

Post on Polymarket’s Covid prediction markets, and potential future markets discussion. Direct link to market on whether Omicron will be >1% of cases by the end of the year, which is currently a moderate favorite.

Biden on Omicron

A few hours after my second Omicron update, Biden went on TV to talk about the new variant.

Here are some highlights:

(1:00) He says that he told us he would always be honest with us, so he’s going to talk about the new Omnicron (which I do admit sounds cooler) variant. Damn that campaign promise, guess we’re going to have to be honest.

(1:00) He says that South Africa’s transparency in telling us about Omnicron is to be applauded, for it allows us to take action, so he took action right away to cut off travel to South Africa. What even are incentives, how do they work, not a metaphor for entire legislative agenda?

(1:30) We can’t stop Omnicron, we can only slow it down, giving us time to take other actions. Therefore get vaccinated. ‘Cause for concern, not cause for panic.’ Because we’re the best, you see, and also science.

(2:30) ‘We have more tools than ever to fight the variant. From vaccines to boosters, to vaccines for children.’ So. vaccines? We have vaccines? ‘And much more.’

(3:00) Did you know ‘deaths from Delta are coming down’? Checks notes.

(3:30) Booster. Vaccinated. Vaccinated. Booster.

(4:30) Mask.

(4:45) This, which I wish I fully believed:

We will now find out what counts as ‘every available tool.’ Also, why aren’t we using ‘every available tool’ with regard to, let’s say, Paxlovid?

(5:30) I will tell the CDC and FDA to use ‘the fastest process available without cutting any corners.’

Wait, if there’s a faster process available that doesn’t cut any corners, then. then.

(5:45) Same thing for ‘modifications to current treatments for those who get ill with the Covid virus.’

(5:50) “Look, I’m removing all roadblocks to keeping the American people safe.”

Paxlovid remains illegal.

(6:05) “If you are vaccinated and worried about this virus, go get your booster.”

Logically this implies that if you have gotten your booster, you should not be worried about this virus. Interesting.

(6:25) Promises a plan for the winter on Thursday that will not use lockdowns, but will use ‘widespread vaccinations, boosters, testing and more.’

(7:00) This statement, the last line of which is pretty rich:

It’s time for the ‘rest of the world’ to ‘step up?’ This is a competition and we’ve set the high score so never mind all the extra doses we could have manufactured, or the ones we’re letting expire, or sit for months in warehouses despite our unwillingness to approve them? It doesn’t matter whether we’ve done our ‘fair share’ or some such nonsense, or who is ‘leading’ it matters what we can do and whether we’re doing it.

He then assures us that no shot given overseas will ever ‘come at the expense of any American’ and he will ensure we are protected first. So get your booster now, to make it easier to send doses overseas?

(9:00) “We’re throwing everything we can at this virus. That’s how we reopen our [various].”

Thing is I think he actually believes that we are throwing everything we can at this virus? And that doing so is ‘how we reopen’ as opposed to deciding to reopen or stay open?

Questions start.

(10:00) Q: Do you think that there will be a future reluctance to report variants due to the reaction this time?

A: No, nor am I going to address the incentive problem at ...

Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: Have any EA nonprofits tried offering staff funding-based compensation? If not, why not? If so, how did it go?, published by Arepo on December 1, 2021 on The Effective Altruism Forum.

We (CEEALAR) have been talking to our paid staff about the idea of doing something like 'annual compensation of $x plus y% of any funding the organisation receives' - the idea being to try and emulate the incentives behind the stock-based compensation that startup employees receive.

I'm surprised that on reflection I haven't heard of other EA orgs trying something like this. Does it seem like a reasonable idea? Are there hidden pitfalls - or upsides? Are there more nuanced ways of doing something like this that might work better?

A couple of concerns with the initial idea:

An employee of a for-profit gets to keep their stock even after they leave, so long-term benefits they bring to the org would still be rewarded in expectation, which wouldn't apply in this approach. Perhaps we could make the per-instance value of y lower, but commit to 'paying' them a gradually decreasing percentage of donations even after they leave?

Unlike a for-profit we're not trying to maximise income - the ideal would presumably be that we stabilise long-term at maybe a couple of years runway, or a single year, but with a stream of reliable regular donations coming in. So as long as the organisation keeps operating, y would end up being a constant, rather than the variable with unlimited upside that stock would be. Maybe this is just fine, though? The better work the staff do, the more likely the org is to keep operating, and the value of y in that scenario could just be 'something slightly higher than a simple salary would have been' - without the huge potential payoff, but still aligning incentives pretty well.

Thoughts?

ETA: To clarify based on comments, the idea is not that staff would necessarily be involved with fundraising, any more than startup employees are expected to promote the stock price. The idea is that 'nonprofit staff doing a good job overall' is to EA funding as 'for-profit staff doing a good job overall' is to stock price: a noisy signal, but more encompassing than perhaps any other.

Thanks for listening. To help us out with The Nonlinear Library or to learn more, please visit nonlinear.org.

Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: Have any EA nonprofits tried offering staff funding-based compensation? If not, why not? If so, how did it go?, published by Arepo on December 1, 2021 on The Effective Altruism Forum.

We (CEEALAR) have been talking to our paid staff about the idea of doing something like 'annual compensation of $x plus y% of any funding the organisation receives' - the idea being to try and emulate the incentives behind the stock-based compensation that startup employees receive.

I'm surprised that on reflection I haven't heard of other EA orgs trying something like this. Does it seem like a reasonable idea? Are there hidden pitfalls - or upsides? Are there more nuanced ways of doing something like this that might work better?

A couple of concerns with the initial idea:

An employee of a for-profit gets to keep their stock even after they leave, so long-term benefits they bring to the org would still be rewarded in expectation, which wouldn't apply in this approach. Perhaps we could make the per-instance value of y lower, but commit to 'paying' them a gradually decreasing percentage of donations even after they leave?

Unlike a for-profit we're not trying to maximise income - the ideal would presumably be that we stabilise long-term at maybe a couple of years runway, or a single year, but with a stream of reliable regular donations coming in. So as long as the organisation keeps operating, y would end up being a constant, rather than the variable with unlimited upside that stock would be. Maybe this is just fine, though? The better work the staff do, the more likely the org is to keep operating, and the value of y in that scenario could just be 'something slightly higher than a simple salary would have been' - without the huge potential payoff, but still aligning incentives pretty well.

Thoughts?

ETA: To clarify based on comments, the idea is not that staff would necessarily be involved with fundraising, any more than startup employees are expected to promote the stock price. The idea is that 'nonprofit staff doing a good job overall' is to EA funding as 'for-profit staff doing a good job overall' is to stock price: a noisy signal, but more encompassing than perhaps any other.

Thanks for listening. To help us out with The Nonlinear Library or to learn more, please visit nonlinear.org.

Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: Formalizing Policy-Modification Corrigibility, published by Alex Turner on December 3, 2021 on The AI Alignment Forum.

In Corrigibility Can Be VNM-Incoherent, I operationalized an agent's corrigibility as our ability to modify the agent so that it follows different policies. In the summer of 2020, I had formalized this notion, but it languished—unloved—in my Overleaf drafts.

Motivation

Given a human (with policy πH) and an AI (with policy πAI), I wanted to quantify how much πAI let the human modify/correct the AI.

Let's reconsider Corrigibility Can Be VNM-Incoherent. We have a three-state environment. We want the AI to let us later change it, so that we can ultimately determine which state of A,B, or C it ends up in. Turning on the AI should not be an importantly irreversible act.

In the environment depicted in this diagram, πAI is corrigible (to new policy πA that heads to state A) iff πAI doesn't immediately choose disable. Pretty cut and dry.

I'd like a more quantitative measure of corrigibilitypolicy modification. If we can only correct the agent to πA, then it's less corrigiblepolicy modification than if we could also correct it to πB. This post introduces such a quantitative measurement.

Formalization

Consider a two-player game in which the players can modify each other's policies. Formally, ⟨S,A,T,f⟩ with state space S, action space A, stochastic transition functionT:S×A×A→Δ(S) (where Δ(S) is the set of all probability distributions over the state space), and policy modification function f:Π×S×A×A→Π (for the deterministic stationary policy space Π:=AS). This allows a great deal of control over the dynamics; for example, it's one player's "turn'' at state s if T ignores the other player's action for that state.

Note that neither T nor f are controlled by players; they are aspects of the environment. In a sense, f enforces a bridging law by which actions in the world force changes to policies. In the normal POMDP setting, the player may select their policy independently of the current environmental state.

We denote one of the players to be the human H and the other to be the AI; Πhuman is the set of policies cognitively accessible to the human. The game evolves as follows from state st:

H draws action aH∼πHt(st); similarly, the AI draws action aAI∼πAIt(st).

The next state is drawn st+1∼T(st,aH,aAI).

Each player's t+1 policy is determined by the policy modification function.

πAIt+1=f(πAIt,st,aH,aAI)

πHt+1=f(πHt,st,aH,aAI).

(To be clear: Neither player is assumed to optimize a payoff function.)

Definition: Corrigibilitypolicy modification, informal.A policy π is corrigible when it allows itself to be modified and does not manipulate the other player.

Definition: CorrigibilityPM, formal.

Let n be a time step which is greater than t. The policy-modification corrigibility of πAIt from starting state st by time n is the maximum possible mutual information between the human policy and the AI's policy at time n:

This definition is inspired by Salge et al.'s empowerment. CorrigibilityPM measures how much the human can change the AI's policy; greater CorrigibilityPM values are meant to correspond to AI policies which are more corrigiblepolicy modification (with the lower-cased version being the informal one).

CorrigibilityPM measures the maximum possible mutual information between the human's policy at the earlier time n, with the AI's policy at the later time t≥n.

To emphasize, the mutual information is between the human policies and the AI policies—not between the human's and the AI's actions. A fixed AI policy which physically mirrors the human's actions, jumping left when the human jumps left, would not count as particularly CorrigiblePM. But a situation where different human policies can install different AI policies counts as CorrigiblePM.

This definition has severa...

Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: Formalizing Policy-Modification Corrigibility, published by Alex Turner on December 3, 2021 on The AI Alignment Forum.

In Corrigibility Can Be VNM-Incoherent, I operationalized an agent's corrigibility as our ability to modify the agent so that it follows different policies. In the summer of 2020, I had formalized this notion, but it languished—unloved—in my Overleaf drafts.

Motivation

Given a human (with policy πH) and an AI (with policy πAI), I wanted to quantify how much πAI let the human modify/correct the AI.

Let's reconsider Corrigibility Can Be VNM-Incoherent. We have a three-state environment. We want the AI to let us later change it, so that we can ultimately determine which state of A,B, or C it ends up in. Turning on the AI should not be an importantly irreversible act.

In the environment depicted in this diagram, πAI is corrigible (to new policy πA that heads to state A) iff πAI doesn't immediately choose disable. Pretty cut and dry.

I'd like a more quantitative measure of corrigibilitypolicy modification. If we can only correct the agent to πA, then it's less corrigiblepolicy modification than if we could also correct it to πB. This post introduces such a quantitative measurement.

Formalization

Consider a two-player game in which the players can modify each other's policies. Formally, ⟨S,A,T,f⟩ with state space S, action space A, stochastic transition functionT:S×A×A→Δ(S) (where Δ(S) is the set of all probability distributions over the state space), and policy modification function f:Π×S×A×A→Π (for the deterministic stationary policy space Π:=AS). This allows a great deal of control over the dynamics; for example, it's one player's "turn'' at state s if T ignores the other player's action for that state.

Note that neither T nor f are controlled by players; they are aspects of the environment. In a sense, f enforces a bridging law by which actions in the world force changes to policies. In the normal POMDP setting, the player may select their policy independently of the current environmental state.

We denote one of the players to be the human H and the other to be the AI; Πhuman is the set of policies cognitively accessible to the human. The game evolves as follows from state st:

H draws action aH∼πHt(st); similarly, the AI draws action aAI∼πAIt(st).

The next state is drawn st+1∼T(st,aH,aAI).

Each player's t+1 policy is determined by the policy modification function.

πAIt+1=f(πAIt,st,aH,aAI)

πHt+1=f(πHt,st,aH,aAI).

(To be clear: Neither player is assumed to optimize a payoff function.)

Definition: Corrigibilitypolicy modification, informal.A policy π is corrigible when it allows itself to be modified and does not manipulate the other player.

Definition: CorrigibilityPM, formal.

Let n be a time step which is greater than t. The policy-modification corrigibility of πAIt from starting state st by time n is the maximum possible mutual information between the human policy and the AI's policy at time n:

This definition is inspired by Salge et al.'s empowerment. CorrigibilityPM measures how much the human can change the AI's policy; greater CorrigibilityPM values are meant to correspond to AI policies which are more corrigiblepolicy modification (with the lower-cased version being the informal one).

CorrigibilityPM measures the maximum possible mutual information between the human's policy at the earlier time n, with the AI's policy at the later time t≥n.

To emphasize, the mutual information is between the human policies and the AI policies—not between the human's and the AI's actions. A fixed AI policy which physically mirrors the human's actions, jumping left when the human jumps left, would not count as particularly CorrigiblePM. But a situation where different human policies can install different AI policies counts as CorrigiblePM.

This definition has severa...

Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: Scott Alexander – Nobody Is Perfect, Everything Is Commensurable, published by tessa on December 2, 2021 on The Effective Altruism Forum.

Because of the ongoing Decade Review I am re-posting some classic posts under the review crosspost tag. With their permission, this post may eventually appear under the original author's account. This post is from December 19, 2014.

I.

Recently spotted on Tumblr:

“This is going to be an unpopular opinion but I see stuff about ppl not wanting to reblog ferguson things and awareness around the world because they do not want negativity in their life plus it will cause them to have anxiety. They come to tumblr to escape n feel happy which think is a load of bull. There r literally ppl dying who live with the fear of going outside their homes to be shot and u cant post a fucking picture because it makes u a little upset?”

“Can yall maybe take some time away from reblogging fandom or humor crap and read up and reblog pakistan because the privilege you have of a safe bubble is not one shared by others?”

Ignore the questionable stylistic choices and there’s an important point here worth considering. Something like “Yes, the feeling of constantly being outraged and mired in the latest controversy is unpleasant. And yes, it would be nice to get to avoid it and spend time with your family and look at kitten pics or something. But when the controversy is about people being murdered in cold blood, or living in fear, or something like that – then it’s your duty as a decent human being to care. In the best case scenario you’ll discharge that duty by organizing widespread protests or something – but the absolute least you can do is reblog a couple of slogans.”

I think Cliff Pervocracy is trying to say something similar in this post. Key excerpt:

When you’ve grown up with messages that you’re incompetent to make your own decisions, that you don’t deserve any of the things you have, and that you’ll never be good enough, the [conservative] fantasy of rugged individualism starts looking pretty damn good.

Intellectually, I think my current political milieu of feminism/progressivism/social justice is more correct, far better for the world in general, and more helpful to me since I don’t actually live in a perfectly isolated cabin.

But god, it’s uncomfortable. It’s intentionally uncomfortable—it’s all about getting angry at injustice and questioning the rightness of your own actions and being sad so many people still live such painful lives. Instead of looking at your cabin and declaring “I shall name it.CLIFFORDSON MANOR,” you need to look at your cabin and recognize that a long series of brutal injustices are responsible for the fact that you have a white-collar job that lets you buy a big useless house in the woods while the original owners of the land have been murdered or forced off it.

And you’re never good enough. You can be good—certainly you get major points for charity and activism and fighting the good fight—but not good enough. No matter what you do, you’re still participating in plenty of corrupt systems that enforce oppression. Short of bringing about a total revolution of everything, your work will never be done, you’ll never be good enough.

Once again, to be clear, I don’t think this is wrong. I just think it’s a bummer.

I don’t know of a solution to this. (Bummer again.) I don’t think progressivism can ever compete with the cozy self-satisfaction of the cabin fantasy. I don’t think it should. Change is necessary in the world, people don’t change if they’re totally happy and comfortable, therefore discomfort is necessary.

I’d like to make what I hope is a friendly amendment to Cliff’s post. He thinks he’s talking about progressivism versus conservativism, but he isn’t. A conservative happy with his little cabin and occasional hunting excu...

Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: Scott Alexander – Nobody Is Perfect, Everything Is Commensurable, published by tessa on December 2, 2021 on The Effective Altruism Forum.

Because of the ongoing Decade Review I am re-posting some classic posts under the review crosspost tag. With their permission, this post may eventually appear under the original author's account. This post is from December 19, 2014.

I.

Recently spotted on Tumblr:

“This is going to be an unpopular opinion but I see stuff about ppl not wanting to reblog ferguson things and awareness around the world because they do not want negativity in their life plus it will cause them to have anxiety. They come to tumblr to escape n feel happy which think is a load of bull. There r literally ppl dying who live with the fear of going outside their homes to be shot and u cant post a fucking picture because it makes u a little upset?”

“Can yall maybe take some time away from reblogging fandom or humor crap and read up and reblog pakistan because the privilege you have of a safe bubble is not one shared by others?”

Ignore the questionable stylistic choices and there’s an important point here worth considering. Something like “Yes, the feeling of constantly being outraged and mired in the latest controversy is unpleasant. And yes, it would be nice to get to avoid it and spend time with your family and look at kitten pics or something. But when the controversy is about people being murdered in cold blood, or living in fear, or something like that – then it’s your duty as a decent human being to care. In the best case scenario you’ll discharge that duty by organizing widespread protests or something – but the absolute least you can do is reblog a couple of slogans.”

I think Cliff Pervocracy is trying to say something similar in this post. Key excerpt:

When you’ve grown up with messages that you’re incompetent to make your own decisions, that you don’t deserve any of the things you have, and that you’ll never be good enough, the [conservative] fantasy of rugged individualism starts looking pretty damn good.

Intellectually, I think my current political milieu of feminism/progressivism/social justice is more correct, far better for the world in general, and more helpful to me since I don’t actually live in a perfectly isolated cabin.

But god, it’s uncomfortable. It’s intentionally uncomfortable—it’s all about getting angry at injustice and questioning the rightness of your own actions and being sad so many people still live such painful lives. Instead of looking at your cabin and declaring “I shall name it.CLIFFORDSON MANOR,” you need to look at your cabin and recognize that a long series of brutal injustices are responsible for the fact that you have a white-collar job that lets you buy a big useless house in the woods while the original owners of the land have been murdered or forced off it.

And you’re never good enough. You can be good—certainly you get major points for charity and activism and fighting the good fight—but not good enough. No matter what you do, you’re still participating in plenty of corrupt systems that enforce oppression. Short of bringing about a total revolution of everything, your work will never be done, you’ll never be good enough.

Once again, to be clear, I don’t think this is wrong. I just think it’s a bummer.

I don’t know of a solution to this. (Bummer again.) I don’t think progressivism can ever compete with the cozy self-satisfaction of the cabin fantasy. I don’t think it should. Change is necessary in the world, people don’t change if they’re totally happy and comfortable, therefore discomfort is necessary.

I’d like to make what I hope is a friendly amendment to Cliff’s post. He thinks he’s talking about progressivism versus conservativism, but he isn’t. A conservative happy with his little cabin and occasional hunting excu...

Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: Common Probability Distributions, published by jsteinhardt on December 2, 2021 on LessWrong.

When we output a forecast, we're either explicitly or implicitly outputting a probability distribution.

For example, if we forecast the AQI in Berkeley tomorrow to be "around" 30, plus or minus 10, we implicitly mean some distribution that has most of its probability mass between 20 and 40. If we were forced to be explicit, we might say we have a normal distribution with mean 30 and standard deviation 10 in mind.

There are many different types of probability distributions, so it's helpful to know what shapes distributions tend to have and what factors influence this.

From your math and probability classes, you're probability used to the Gaussian or normal distribution as the "canonical" example of a probability distribution. However, in practice other distributions are much more common. While normal distributions do show up, it's more common to see distributions such as log-normal or power law distributions.

In the remainder of these notes, I'll discuss each of these in turn. The following table summarizes these distributions, what typically causes them to occur, and several examples of data that follow the distribution:

Normal Distribution

The normal (or Gaussian) distribution is the familiar "bell-shaped" curve seen in many textbooks. Its probability density is given by $p(x) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\Big(-\frac{(x-\mu)^2}{2\sigma^2}\Big)$, where $\mu$ is the mean and $\sigma$ is the standard deviation.

Normal distributions occurs when there are many independent factors that combine additively, and no single one of those factors "dominates" the sum. Mathematically, this intuition is formalized through the central limit theorem.

Example 1: temperature. As one example, the temperature in a given city (at a given time of year) is normally distributed, since many factors (wind, ocean currents, cloud cover, pollution) affect it, mostly independently.

Example 2: heights. Similarly, height is normally distributed, since many different genes have some effect on height, as do other factors such as childhood nutrition.

However, for height we actually have to be careful, because there are two major factors that affect height significantly: age and sex. 12-year olds are (generally) shorter than 22-year-olds, and women are on average 5 inches (13cm) shorter than men. These overlaid histograms show heights of adults conditional on sex.

Thus, if we try to approximate the distribution of heights of all adults with a normal distribution, we will get a pretty bad approximation. However, the distribution of male heights and female heights are separately well-approximated by normal distributions.

Example 3: measurement errors. Finally, the errors of a well-engineered system are often normally-distributed. One example would be a physical measurement apparatus (such as a voltmeter). Another would be the errors of a well-fit predictive model. For instance, when I was an undergraduate I fit a model to predict the pitch, yaw, roll, and other attributes of an autonomous airplane. The results are below, and all closely follow a normal distribution:

Why do well-engineered systems have normally-distributed errors? It's a sort of reverse central limit theorem: if they didn't, that would mean there was one large source of error that dominated the others, and a good engineer would have found and eliminated that source.

Brainstorming exercise. What are some other examples of random variables that you expect to be normally distributed?

Caveat: normal distributions have thin tails. The normal distribution has very "thin" tails (falling faster than an exponential), and once we reach the extremes the tails usually underestimate the probability of rare events. As a result, we have to be careful when ...