Here’s a fun question – let’s consider, purely in the abstract, the notion of quickly putting on a lot of weight. If, hypothetically, one were to weigh themselves every day and, conceptually, throw away all of the results which showed an increase in weight, what, in a strictly Platonic sense, are the odds that actually they’re just a fat shit not actually losing any weight?

**Simulation**

Let’s engage in this fun thought experiment, as ridiculous as it may seem to the better amongst us who eat well and exercise regularly.

Suppose your real weight fluctuates about a mean with a standard deviation . Then, when measuring your weight on any given day , suppose the probability density of measuring a weight is distributed normally as:

i.e. independent of – there is no overall trend to your weight.

Also suppose that you keep track of your weight, and only record a measurement when it is below your previous record low weight. A sequence of measurements may then look like this, monotonically decreasing:

Here we’ve simulated the measurement process – every day, draw a sample from , and if it is low enough, accept it and plot it. It looks great – this hypothetical human has lost several kilograms in just a few months!

It is, sadly, a lie. Here it is overlaid on many similar simulations and their average:

The trend is clearly a rapid initial ‘fall’, followed by an increasingly slow decline – intriguingly similar to the expected weight-loss profile if you were actually caring about your body.

Let’s think about how to model this process in a couple of ways.

**Method 1: Iterated expectation**

Starting from the beginning, on day 1 the expected value of the measured weight is just (handwaving the integral limits away):

Easy enough. What about day 2? This is more interesting – the probability of the actual weight stays the same, but not the reported measurement – the latter is clamped at a maximum of :

The particularly eager reader can confirm that this evaluates to

where the error function is defined as an integral of the gaussian.

Each day’s measurement is then a function of the previous days – hence ‘iterated expectation’. Let’s see what this looks like compared to the simulation:

It’s not terrible, but it’s not perfect either. The general shape is right, and it might converge to a similar value, but we can do better.

**Method 2: Expected minimum**

There is another, simpler, way to look at this problem: after days, the average weight recorded will just be the *minimum *of all of the measurements so far. Fortunately, there’s a generic way to calculate the distribution of this minimum.

As the distribution of our measurements is constant over time – it exhibits homoscedasticity – for each day is there is a certain probability of measuring a value *under *:

where is known as the cumulative distribution function of . The probability of having a minimum value *over * after days is therefore the probability of all measurements so far being over :

We’re interested in the conjugate case, where which is simply

The probability density distribution of the minimum measured value after days is then the derivative of this expression:

Phew. We’re not quite there – to get the expected value, we need to integrate this expression again. Unfortunately there’s not a closed form (as far as I can tell), but we can still do it numerically:

Bang on! Well done us.

**Method 3:** **Obscure Stack Exchanging**

The final method is to go spelunking into the dusty corners of the internet. This comment suggests that a simple approximation is:

Plotting it out, it isn’t too bad! Points to whoever can derive it for me:

And finally, a comparison of all methods:

…**so?**

Although the last method results in the worst approximation, it probably represents a good rule of thumb for figuring out if you’ve actually lost weight:

- Days 1-10: calculate the mean and standard deviation of your natural weight by measuring it properly every day.
- Days 10-99: diet, don’t diet, who cares – just keep a record of your minimum weight so far. We all know careful record keeping is better for the body than any amount of quinoa anyway.
- Day 100: after 90 days, (yep, 4 nines!), so if you are 3 or more standard deviations below your starting weight, you’ve probably smashed it. Well done! Celebrate by doing some more maths.

Cool post!! I never thought about it haha, I guess the same can be said from the other perspective, ie: you only record your maximum weight instead of the minimum. So maybe I’m not getting fat, it’s just a statistical artefact š

I have a question about your method. Which $\sigma$ did you choose for your distribution? I guess the results depend a lot on that since a bigger sigma would make the illusion of losing/winning weight faster.

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