Zeroth and First Order Forecasting

Lecture 4

Stat 165 Spring 2025

Slides credit: Jacob Steinhardt

Logistics

Logistical Reminders

• HW2 due tomorrow

Imagine a scenario…

• You’re trying to predict how long it will take to finish your HW assignment�
• You think about all the problems and how long each will take

Sure, Q4 looks a little hard, but if I try real hard I can solve it in 2 hours… so maybe 7 hours total?

• You start at 1pm the day it’s due, �and finish at 3am��(...just like the last 3 assignments)

Imagine a scenario…

• You’re getting ready for class in the morning
• Your first class is at 9:10, so when should you leave?

Google maps says 8 minutes, but I’m a �fast biker so 6 should be enough.

• You leave at 9:04, and �get to class at 9:20��(...just like the last several times)

Aside: The Planning Fallacy

• These are both instances of the planning fallacy, where we tend to be over-optimistic about how long our plans will take to finish.�
• A good rule of thumb is to multiply by 2x-3x.�
• An even better rule of thumb is to think of the last several �similar times, and take the average.�

This is called reference class forecasting or zeroth-order approximation, and is widely useful (even beyond planning).

Zeroth-order Approximation

��Assume that things don’t change�(i.e., approximate with constant function).

HW time this week = HW time last week��Commute time today = commute time yesterday

Use Cases of Zeroth-Order Approximation

Avoiding the planning fallacy�

Making a budget�

��� Policy: carbon emissions next year

Brainstorming Exercise

List three other forecasting questions or other applications where a zeroth-order approximation might be useful.

Zeroth-order Consistency

In February 2020, suppose you’re trying to figure out what the world will look like in April 2020. Here are 3 different zeroth-order approximations:

1. Assume the world will stay roughly the same�(i.e., April 2020 will look like February 2020)
2. Assume number of available ICU beds will stay roughly the same
3. Assume number of Covid-19 cases will stay roughly the same

�Most people would have disagreed with assumption 3. But implicitly acted as if assumption 1 was true (even though these were ~the same).

First-order Approximation

� (spoiler alert)�����For Feb. 2020 prediction, a better strategy would be to apply a first-order approximation to the number of Covid cases.

Two possibilities:

• Linear approximation (d/dt tomorrow = d/dt today)
• Log-linear approximation (% growth tomorrow = % growth today)

Linear approximation

Log-linear approximation

Log-linear is better here, but which one to use is generally an important choice!

Brainstorming Question

List at least two settings where a first-order approximation would likely be better than zeroth-order.

​

For each of these, would you use linear or log-space for your prediction?

Breakdowns of First-order Approximation

Most trends have to stop eventually: [trends listed as of Jan. 2024]

• Tesla has grown ~50%/year from 2015 to 2022. If continued, would be >50% of US economy by ~2035.
• Number of compute-hours used in AI experiments grew on average 0.7 orders of magnitude/year from 2012 to 2023. In 6-7 years this would exceed world’s total hardware budget.
• Early Covid growth: double every 4.5 days (4 months to hit world population).
• Saturation effects: fast-growing trends eventually hit limits.

Breakdowns of First-order Approximation

Deportations under Trump��How many deportations do�you predict in 2018?

�(Groups of 2-3, 2 minutes)

Answer

Brainstorming Exercise

What are other places where zeroth-order (and even first-order) might not work well? How could we tell?

Choosing the right first-order approximation

Does first-order approximation work for all 3 COVID metrics from before?

1. What the world looks like overall
2. Number of ICU beds available
3. Number of Covid cases

�Only sometimes! Applying first-order approx to 1. or 2. would not make right prediction. Important to notice and use 3. to make predictions.

Zeroth and First-order Contradictions

1st-order for # covid cases contradicts 0th-order for # of lockdowns (in Feb. 2020)

�1st-order for Tesla stock growth contradicts 0th-order (and 1st-order) for US economy���Question. When a 0th- and 1st-order approximation contradict, what heuristics can help us decide which one to follow?