Upper Bound / Lower Bound Estimation. Or How to Deal with Uncertainty?

When talking about variable bitrate, we encountered a situation where we couldn’t give a precise answer to the question: how much space is needed on an SSD / HDD / “flash drive” to record a two-hour match if the video recording algorithm uses variable bitrate?

Variable bitrate introduces uncertainty into our task. The specific bitrate value during recording will depend on what happens at each moment in front of the camera lens. We cannot look into the future and predict exactly how the game will unfold, how active the stands will be, and so on.

That is, in our case, uncertainty is a fundamental property of the task itself. So, what can we do? Give up and say this task is unsolvable in principle? Or try to find an approach that allows us to answer the practical question: what size disk should we take to ensure the entire match recording definitely fits on it?

Let’s imagine the most dynamic match in history: both teams are all-star squads playing at their limit, constantly attacking, the camera switches quickly, there’s a storm in the stands. What would be the maximum bitrate? By multiplying this bitrate by the duration, we get the maximum possible file size. If our disk (or flash drive) can hold that much, then our match will definitely fit. This is an upper bound estimate.

Now let’s imagine the match was postponed due to fog. The camera films an empty field in thick fog for 2 hours (obviously, no one would actually do this, and we’re using this example only to understand the principle itself). The file size will be minimal. This is a lower bound estimate.

Throughout this example, we assume that the encoder settings, resolution, frame rate, audio settings, and all other recording parameters are fixed. Under this assumption, the dynamic match and the foggy empty field are treated as the highest- and lowest-bitrate scenarios, respectively.

Let’s consider another example of a lower bound estimate. Suppose you want to buy an apartment that costs $1\,\text{million dollars}$ .

You earn $10\,\text{thousand dollars per month}$ . How many months do you need to save money to accumulate $1\,\text{million dollars}$ ? Off the top of your head, it might be difficult to answer because you don’t yet have a clear idea of exactly how much you can save each month. But you can instantly produce a lower bound estimate—calculate the minimum number of months it would take.

Assume you save all the money you earn (spend nothing on food, rent, transport, phone, etc.)—that is, all $10\,\text{thousand dollars}$ . How many months would it then take to save $1\,\text{million dollars}$ ?

Divide $1{,}000{,}000$ by $10{,}000$ , you get $100\,\text{months}$ . Or $8\,\text{years}$ and $4\,\text{months}$ .

There’s your lower bound estimate.

Note

In reality, of course, everything is much more complicated—apartment prices can rise or fall, money can depreciate, your salary might increase over time, or decrease if you have to switch to a lower-paying job. But regardless, a lower bound estimate provides a reference point for further reasoning and calculations.

Addition

There are also more complex statistical approaches to dealing with uncertainty that seek not an absolute minimum or maximum, but a typical or expected value.

For example, one could analyze hundreds of recorded football matches and derive the average bitrate. But that is a topic for a separate, extensive discussion about probability and statistics, which we will return to someday.

For the Curious

The cat in the picture refers to the famous thought experiment in quantum mechanics—Schrödinger’s Cat—which illustrates the quantum measurement problem. Here we use this image only as a playful metaphor for uncertainty: the impossibility of predicting a match’s bitrate in advance is, of course, not quantum in nature.