Earlier today I was enjoying Andrew Charlton’s Probabilistic Thinking in SEO post when I came across something that puzzled me.
Andrew is describing how to combine a probability range (e.g. “we think this will be somewhere between 5% and 60%”) into a single probability. He uses the geometric mean for this:
You can convert your bounds into a point estimate. Taking ‘the average’ will likely be well off, but you can use something called the geometric mean instead. The geometric mean is better suited to data with outliers or extreme variance.
I couldn’t figure out why it was best to use the geometric mean rather than the arithmetic (normal) mean. So I pinged him a direct message to ask.
Andrew kindly explained to me that the geometric mean is useful when you are more interested in ending up in the middle of something that spans several orders of magnitude. For example the arithmetic mean of 1 and 1 million is close enough to 500,000 which is one order of magnitude smaller than 1 million but five orders of magnitude bigger than 1. So in one sense this average is “closer” to 1 million than it is to one.
The geometric mean of these two numbers is
sqrt(1*1000000)=1000 which is in the middle in terms of orders of magnitude. And since Andrew is using these numbers to make a Fermi estimate where he aims to be within an order of magnitude of the true value this approach makes perfect sense.
This tweet is where it finally clicked for me:
I see the order of magnitude estimate as “really” an estimate of the log, so GM of estimates makes sense as it corresponds to AM of logs. Still trying to justify the AGM though :-)— Robert Low (@RobJLow) November 13, 2017
But this got me thinking about probabilities and how we average them. Orders of magnitude are a bit different for probabilities because all probabilities must be between zero and one. It is very similar at the low end where 0.1, 0.01, 0.001 and 0.0001 are all different orders of magnitude but the same is true at the high end too when you look at 0.99, 0.999 and 0.9999.
Consider a service with 99% uptime; it can be down for nearly four days per year.
With 99.9% there must be less than 9 hours downtime per year.
99.99% requires less than 1 hour and a 99.999% SLA requires a maximum of slightly more than 5 minutes of downtime per year.
Normally when dealing with probabilities this kind of problem is dealt with by using a logit transform. We can use this to calculate the “logistic mean” in the same way that a log transform gives us the geometric mean.
The calculation looks like this:
Here is the “logistic mean” of 0.9 and 0.999 calculated on Wolfram Alpha. The answer is 0.9896 which is almost exactly 0.99; the middle “order of magnitude” between the inputs.
I haven’t seen this written about anywhere else; probably because I’m not putting the right words for it into Google. But just in case this is something new I thought I’d document it here.
[Andrew has a cool looking SEO Forecasting course that you should have a look at]
Last month was an unusually social one for me with two virtual events.
First was a long term favourite of mine; MeasureCamp. And then a new one (for me); Bhav Patel’s CRAP Talks.
This was my first time attending a virtual MeasureCamp although I knew a bit of what to expect because I had been involved in a test event in lockdown number one.
I presented the following slides to introduce a discussion on “MLUX”; the user experience of working with machine learning.
The discussion that followed didn’t really talk about forecasting at all (which is fine - this is a bigger problem than one area). People expressed frustration with things like product recommendations that lag six months behind the current fashions and the difficulty of comparing the uplift from different ways of doing things.
From one angle it is easy to say that the problem lies in a lack of skills/training for the relevant people. But this will never be a complete solution because there are so few places where one can learn about the development and training of world class models.
In my (biased - look where you are reading this!) opinion improvements in the tooling is more likely to be the best way forward. Libraries like Keras or Huggingface’s transformers allow someone with my level of skill to get started in this area - I would not be able to code or train stuff like this from scratch. And I think similar tooling can and should exist for other people too.
In this case CRAP stands for Conversion Rate, Analytics and Product. I have been aware of it as a offline meetup for a few years now but because it is mainly in London and I don’t have a focus on conversion rate optimisation or product analytics I have never made the effort to attend. This is my loss as I thought the virtual event was excellent.
The main part of the event was a presentation from Facebook’s marketing science team about Robyn which is a new open source project they have launched to help with Marketing Mix Modelling.
I have done a small amount of MMM in the past and I wish Robyn had been available then. I was learning as I went along without the support of someone with more experience so I spent a lot of time thinking and worrying about the details of what I was doing and whether or not it was the right approach. A page like this would have saved me so much stress even if didn’t also come with code that does all the calculations for you.
Following this presentation there was a discussion on forecasting which Bhav was kind enough to invite me to be a part of. I think he was hoping for some juicy arguments but (boringly!) I mostly agreed with everyone else. A quick summary of some of the points covered:
I think I went a bit off piste talking about how your forecast is your “model of the world” but people seemed very tolerant of my ramblings ;-)
I look forward to seeing what else comes out of CRAP Talks in the months and years to come.
Christmas 2020 was on a Friday which meant that Boxing Day was on a Saturday. In the UK Boxing Day is a public holiday but because it fell on a weekend the following Monday (the 28th December) was a “substitute day” public holiday.
The database of country holidays used by Forecast Forge treats these two things as different holidays. i.e. in 2020 there was a Boxing Day holiday and an extra substitute day Boxing Day on the 28th.
The last year with a substitute Boxing Day was 2015 so unless the historical data used in the forecast includes Christmas 2015 then the Forecast Forge algorithm will not have any relevant training data to make a prediction for the holiday effect in 2020.
I tweeted this observation last week and Peter O’Neill replied with this:
Wait - are you saying that machine learning models will all just break on unexpected days?— Peter O'Neill (@peter_oneill) December 28, 2020
Which I think is a very interesting question.
The algorithm is behaving exactly as it is designed to so in one (important!) respect it is not broken at all. But on the other hand if it isn’t doing what the user expects then even if it isn’t fair to describe the procedure as “broken” it is certainly far from ideal.
And this gets to the heart of why I think it is an interesting question; it is about how users interact with machine learning systems and what expectations are reasonable for such an interaction. And this is exactly the area where the user interface for Forecast Forge sits!
Let’s talk specifically about the substitute day bank holiday problem even though there must be about a million other similar issues.
Off the top of my head I can think of six different ways of modelling this:
It isn’t obvious to me that any one of these is always going to be better than all the others.
For a bricks and mortar store I can imagine that, in a non-Covid year, they would get a lot more footfall with a substitute day than in a normal year. For an online brand I can see it not making very much difference - particularly for Boxing Day which lies in the dead period between Christmas and New Year.
The good news is that Forecast Forge allows you to use any of these options by setting up appropriate helper columns. The bad news is that it doesn’t communicate at all about which of the models you are using with the default “country holidays” model.
It is a huge challenge for me to build a user interface that knows when a modelling choice has other options that the user might want to explore. People don’t use Forecast Forge because they want to have to think about this for every possible modelling decision; people like this need to code their own models. But, just as importantly, people do use Forecast Forge because they want a bit more control and customisation rather than a forecasting service where they get the result and can’t do anything about what it says even if it is obviously wrong.
And not knowing what modelling decisions have been made behind the scenes and what other options are possible makes this much harder than I would like for users.
As Peter says:
My argument has always been for a tool that can learn from past behaviour but allow for a human to adjust levers based on known upcoming events using a human estimated weighting— Peter O'Neill (@peter_oneill) December 28, 2020
And this would be the holy grail for human/machine learning interaction. It already exists for those who are confident with both coding and converting whatever levers a business might want to pull into mathematics. But these people are a rare (and expensive!) breed; I wouldn’t class myself as top tier at either discipline.
I have made design decisions with Forecast Forge that do restrict the levers that are available for pulling. My hope is that I have left the most important levers and that by building it into a spreadsheet all my users have more flexibility in what they can do with it than in other tools.
But it is still short of what I wish it could be. Is there a way to hide the unnecessary complexity whilst making all the necessary complexity visible?
Here is how I make forecasts using Forecast Forge. There are a lot of similarities with how forecasting is done by data scientists so some of this will be useful even if you aren’t a Forecast Forge subscriber.
I cannot stress enough how important it is to plot the data! For forecasting problems it isn’t even that complicated; putting the date along the x-axis and the metrics you are interested in on the y-axis will be enough in most cases.
A Covid-19 vaccination is now clearly visible on the horizon but there are still significant uncertainties about when exactly this will become available and what will happen in the meantime.
In the UK I think there is likely to be a period of national lockdown in January. And maybe more in the Spring too. Other countries in Europe are in a similar situation and who knows what will be going on in the USA by then?
It is important for businesses to be able to make good estimates for what their demand will look like at these times. I will show you how to do this in Forecast Forge.