In the grand HN tradition of being triggered by a word in the post and going off on a not-quite-but-basically-totally-tangential rant:
There’s (at least) three areas here that are footguns with these kinds of calculations:
1) 95% is usually a lot wider than people think - people take 95% as “I’m pretty sure it’s this,” whereas it’s really closer to “it’d be really surprising if it were not this” - by and large people keep their mental error bars too close.
2) probability is rarely truly uncorrelated - call this the “Mortgage Derivatives” maxim. In the family example, rent is very likely to be correlated with food costs - so, if rent is high, food costs are also likely to be high. This skews the distribution - modeling with an unweighted uniform distribution will lead to you being surprised at how improbable the actual outcome was.
3) In general normal distributions are rarer than people think - they tend to require some kind of constraining factor on the values to enforce. We see them a bunch in nature because there tends to be negative feedback loops all over the place, but once you leave the relatively tidy garden of Mother Nature for the chaos of human affairs, normal distributions get pretty abnormal.
I like this as a tool, and I like the implementation, I’ve just seen a lot of people pick up statistics for the first time and lose a finger.
I think to do all that you’d need a full on DSL rather than something pocket calculator like. I think adding a triangular distribution would be good though.
This jives with my general reaction to the post, which was that the added complexity and difficulty of reasoning about the ranges actually made me feel less confident in the result of their example calculation. I liked the $50 result, you can tack on a plus or minus range but generally feel like you're about breakeven. On the other hand, "95% sure the real balance will fall into the -$60 to +$220 range" feels like it's creating a false sense of having more concrete information when you've really just added compounding uncertainties at every step (if we don't know that each one is definitely 95%, or the true min/max, we're just adding more guesses to be potentially wrong about). That's why I don't like the Drake equation, every step is just compounding wild-ass guesses, is it really producing a useful number?
It is producing a useful number. As more truly independent terms are added, error grows with the square root while the point estimation grows linearly. In the aggregate, the error makes up less of the point estimation.
This is the reason Fermi estimation works. You can test people on it, and almost universally they get more accurate with this method.
If you got less certain of the result in the example, that's probably a good thing. People are default overconfident with their estimated error bars.
They are meaning the same thing. The original comment pointed out that people’s qualitative description and mental model of the 95% interval means they are overconfident… they think 95 means ‘pretty sure I’m right’ rather than ‘it would be surprising to be wrong’
I strongly agree with this, and particularly point 1. If you ask people to provide estimated ranges for answers that they are 90% confident in, people on average produce roughly 30% confidence intervals instead. Over 90% of people don't even get to 70% confidence intervals.
I don't think estimation errors regarding things outside of someone's area of familiarity say much.
You could ask a much "easier"" question from the same topic area and still get terrible answers: "What percentage of blue whales are blue?" Or just "Are blue whales blue?"
Estimating something often encountered but uncounted seems like a better test. Like how many cars pass in front of my house every day. I could apply arithmetic, soft logic and intuition to that. But that would be a difficult question to grade, given it has no universal answer.
I have no familiarity with blue whales but I would guess they're 1--5 times the mass of lorries, which I guess weigh like 10--20 cars which I in turn estimate at 1.2--2 tonnes, so primitively 12--200 tonnes for a normal blue whale. This also aligns with it being at least twice as large as an elephant, something I estimate at 5 tonnes.
The question asks for the heaviest, which I think cannot be more than three times the normal weight, and probably no less than 1.3. That lands me at 15--600 tonnes using primitive arithmetic. The calculator in OP suggests 40--320.
The real value is apparently 170, but that doesn't really matter. The process of arriving at an interval that is as wide as necessary but no wider is the point.
Estimation is a skill that can be trained. It is a generic skill that does not rely on domain knowledge beyond some common sense.
I did a project with non-technical stakeholders modeling likely completion dates for a big GANTT chart. Business stakeholders wanted probabilistic task completion times because some of the tasks were new and impractical to quantify with fixed times.
Stakeholders really liked specifying work times as t_i ~ PERT(min, mode, max) because it mimics their thinking and handles typical real-world asymmetrical distributions.
[Background: PERT is just a re-parameterized beta distribution that's more user-friendly and intuitive https://rpubs.com/Kraj86186/985700]
This looks like a much more sophisticated version of PERT than I have seen used. When people around me have claimed to use PERT, they have just added together all the small numbers, all the middle numbers, and all the big numbers. That results in a distribution that is too extreme in both lower and upper bound.
The android app fits lognormals, and 90% rather than 95% confidence intervals. I think they are a more parsimonious distribution for doing these kinds of estimates. One hint might be that, per the central limit theorem, sums of independent variables will tend to normals, which means that products will tend to be lognormals, and for the decompositions quick estimates are most useful, multiplications are more common
I have made a similar tool but for the command line[1] with similar but slightly more ambitious motivation[2].
I really like that more people are thinking in these terms. Reasoning about sources of variation is a capability not all people are trained in or develop, but it is increasingly important.[3]
On the whole it seems like a nice idea, but there's a couple of weird things, such as:
> Note: If you're curious why there is a negative number (-5) in the histogram, that's just an inevitable downside of the simplicity of the Unsure Calculator. Without further knowledge, the calculator cannot know that a negative number is impossible (in other words, you can't have -5 civilizations, for example).
The input to this was "1.5~3 x 0.9~1.0 x 0.1~0.4 x 0.1~1.0 x 0.1~1.0 x 0.1~0.2 x 304~10000" - every single range was positive, so regardless of what this represents, it should be impossible to get a negative result.
I guess this is a consequence of "I am not sure about the exact number here, but I am 95% sure it's somewhere in this range" so it's actually considering values outside of the specified range. In this case, 10% either side of all the ranges is positive except the large "304~10000".
Trying with a simpler example: "1~2 x 1~2" produces "1.3~3.4" as a result, even though "1~4" seems more intuitive. I assume this is because the confidence of 1 or 4 is now only 90% if 1~2 was at 95%, but it still feels off.
I wonder if the 95% thing actually makes sense, but I'm not especially good at stats, certainly not enough to be sure how viable this kind of calculator is with a tighter range. But just personally, I'd expect "1~2" to mean "I'm obviously not 100% sure, or else I wouldn't be using this calculator, but for this experiment assume that the range is definitely within 1~2, I just don't know where exactly".
>The input to this was "1.5~3 x 0.9~1.0 x 0.1~0.4 x 0.1~1.0 x 0.1~1.0 x 0.1~0.2 x 304~10000" - every single range was positive, so regardless of what this represents, it should be impossible to get a negative result.
Every single range here includes positive and negative numbers. To get the correct resulting distribution you have to take into account the entire input distribution. All normal distributions have a non-zero possibility to be negative.
If you want to consider only the numbers inside the range you can look at interval arithmetic, but that does not give you a resulting distribution.
The calculator in Emacs has support for what it is you request, which it calls "interval forms". Interval form arithmetic simply means executing the operations in parallel on both ends of the interval.
It also has support for "error forms" which is close to what the calculator in OP uses. That takes a little more sophistication than just performing operations on the lower and upper number in parallel. In particular, the given points don't represent actual endpoints on a distribution, but rather low and high probability events. Things more or less likely than those can happen, it's just rare.
> I'm not especially good at stats
It shows! All the things you complain about make perfect sense given a little more background knowledge.
Is it actually just doing it at both ends or something nore complex? Because for example if I did 7 - (-1~2)^2 the actual range would be 3-7 but just doing both ends of the interval would give 3-6 as the function is maximised inside the range.
> every single range was positive, so regardless of what this represents, it should be impossible to get a negative result.
They explain that the range you give as input is seen as only being 95% correct, so the calculator adds low-probability values outside of the ranges you specified.
I can see how that surprises you, but it's also a defensible design choice.
The ASCII art (well technically ANSI art) histogram is neat. Cool hack to get something done quickly. I'd have spent 5x the time trying various chart libraries and giving up.
Would be nice to retransform the output into an interval / gaussian distribution
Note: If you're curious why there is a negative number (-5) in the histogram, that's just an inevitable downside of the simplicity of the Unsure Calculator. Without further knowledge, the calculator cannot know that a negative number is impossible
Drake Equation or equation multiplying probabilities can also be seen in log space, where the uncertainty is on the scale of each probability, and the final probability is the product of exponential of the log probabilities. And we wouldnt have this negative issue
I think arbitrary distribution choice is dangerous. You're bound to end up using lots of quantities that are integers, or positive only (for example). "Confidence" will be very difficult to interpret.
Does it support constraints on solutions? E.g. A = 3~10, B = 4 - A, B > 0
It sounds like a gimmick at first, but looks surprisingly useful. I'd surely install it if it was available as an app to use alongside my usual calculator, and while I cannot quite recall a situation when I needed it, it seems very plausible that I'll start finding use cases once I have it bound to some hotkey on my keyboard.
This is neat! If you enjoy the write up, you might be interested in the paper “Dissolving the Fermi Paradox” which goes even more on-depth into actually multiplying the probability density functions instead of the common point estimates. It has the somewhat surprising result that we may just be alone.
I think they should be functions: G(50, 1) for a Gaussian with µ=50, σ=1; N(3) for a negative exponential with λ=3, U(0, 1) for a uniform distribution between 0 and 1, UI(1, 6) for an uniform integer distribution from 1 to 6, etc. Seems much more flexible, and easier to remember.
Love it! I too have been toying with reasoning about uncertainty. I took a much less creative approach though and just ran a bunch of geometric brownian motion simulations for my personal finances [0]. My approach has some similarity to yours, though much less general. It displays the (un)certainty over time (using percentile curves), which was my main interest. Also, man, the UI, presentation, explanations: you did a great job, pretty inspiring.
I want to ask about adjacent projects - user interface libraries that provide input elements for providing ranges and approximate values. I'm starting my search around https://www.inkandswitch.com/ and https://malleable.systems/catalog/ but I think our collective memory has seen more examples.
This is awesome. I used Causal years ago to do something similar, with perhaps slightly more complex modelling, and it was great. Unfortunately the product was targeted at high paying enterprise customers and seems to have pivoted into finance now, I've been looking for something similar ever since. This probably solves at least, err... 40~60% of my needs ;)
https://qalculate.github.io can do this also for as long as I've used it (only a couple years to be fair). I've got it on my phone, my laptop, even my server with apt install qalc. Super convenient, supports everything from unit conversion to uncertainty tracking
The histogram is neat, I don't think qalc has that. On the other hand, it took 8 seconds to calculate the default (exceedingly trivial) example. Is that JavaScript, or is the server currently very busy?
It's all computed in the browser so yeah, it's JavaScript. Still, 8 seconds is a lot -- I was targeting sub-second computation times (which I find alright).
I didn't peruse the source code. I just read the linked article in its entirety and it says
> The computation is quite slow. In order to stay as flexible as possible, I'm using the Monte Carlo method. Which means the calculator is running about 250K AST-based computations for every calculation you put forth.
So therefore I conclude Monte Carlo is being used.
Line 19 to 21 should be the Monte-Carlo sampling algorithm. The implementation is maybe a bit unintuitive but apparently he creates a function from the expression in the calculator, calling that function gives a random value from that function.
Interesting. I like the notation and the histogram that comes out with the output. I also like the practical examples you gave (e.g. the application of the calculator to business and marketing cases). I will try it out with simple estimates in my marketing campaigns.
Very cool. This can also be used for LLM cost estimation. Basically any cost estimation I suppose. I use cloudflare workers a lot and have a few workers running for a variable amount of time. This could be useful to calculate a ball park figure of my infra cost. Thank you!
I'm guessing this is not an error. If you divide 1/normal(0,1), the full distribution would range from -inf to inf, but the 95% output doesn't have to.
I don't quite understand, probably because my math isn't good enough.
If you're treating -1~1 as a normal distribution, then it's centered on 0. If you're working out the answer using a Monte Carlo simulation, then you're going to be testing out different values from that distribution, right? And aren't you going to be more likely to test values closer to 0? So surely the most likely outputs should be far from 0, right?
When I look at the histogram it creates, it varies by run, but the most common output seems generally closest to zero (and sometimes is exactly zero). Wouldn't that mean that it's most frequently picking values closest to -1 or 1 denoninator?
OK, but do we necessarily just care about the central 95% range of the output? This calculation has the weird property that values in the tails of the input correspond to values in the middle of the output, and vice versa. If you follow the intuition that the range you specify in the input corresponds to the values you expect to see, the corresponding outputs would really include -inf and inf.
Now I'm realizing that this doesn't actually work, and even in more typical calculations the input values that produce the central 95% of the output are not necessarily drawn from the 95% CIs of the inputs. Which is fine and makes sense, but this example makes it very obvious how arbitrary it is to just drop the lowermost and uppermost 2.5%s rather than choosing any other 95/5 partition of the probability mass.
That may be true, but if you look at the distribution it puts out for this, it definitely smells funny. It looks like a very steep normal distribution, centered at 0 (ish). Seems like it should have two peaks? But maybe those are just getting compressed into one because of resolution of buckets?
”Without further knowledge, the calculator cannot know that a negative number is impossible (in other words, you can't have -5 civilizations, for example).”
Not true. If there are no negative terms, the equation cannot have negative values.
The calculator cannot know whether there are no negative terms. For example, if people's net worth is distributed 0.2–400, there's likely a significant chunk of people who are, on the whole, in debt. These will be represented as a negative term, even though their distribution was characterised by positive numbers.
The range notation indicates 95% confidence intervals, not the minima and maxima. If the lower bounds are close enough to zero (and the interval is large enough), then there may some residual probability mass associated with negative values of the variable.
I actually stumbled upon this a while ago from social media and the web version has a somewhat annoying latency, so I wrote my own version in Python. It uses numpy so it's faster. https://gist.github.com/kccqzy/d3fa7cdb064e03b16acfbefb76645... Thank you filiph for this brilliant idea!
The reason I'm asking: unsure also has a CLI version (which is leaps and bounds faster and in some ways easier to use) but I rarely find myself using it. (Nowadays, I use https://filiph.github.io/napkin/, anyway, but it's still a web app rather than a CLI tool.)
Here (https://uncertainty.nist.gov/) is another similar Monte Carlo-style calculator designed by the statisticians at NIST. It is intended for propagating uncertainties in measurements and can handle various different assumed input distributions.
I think I was looking at this and several other similar calculators when creating the linked tool. This is what I mean when I say "you'll want to use something more sophisticated".
The problem with similar tools is that of the very high barrier to entry. This is what my project was trying to address, though imperfectly (the user still needs to understand, at the very least, the concept of probability distributions).
If I am reading this right, a range is expressed as a distance between the minimum and maximum values, and in the Monte Carlo part a number is generated from a uniform distribution within that range[1].
But if I just ask the calculator "1~2" (i.e. just a range without any operators), the histogram shows what looks like a normal distribution centered around 1.5[2].
Shouldn't the histogram be flat if the distribution is uniform?
> Range is always a normal distribution, with the lower number being two standard deviations below the mean, and the upper number two standard deviations above. Nothing fancier is possible, in terms of input probability distributions.
Part of the confusion here is likely that the tool, as seen on the web, probably lags significantly behind the code. I've started using a related but different tool (https://filiph.github.io/napkin/).
The HN mods gave me an opportunity to resubmit the link, so I did. If I had more time, I'd have also upgraded the tool to the latest version and fix the wording. But unfortunately, I didn't find the time to do this.
An alternative approach is using fuzzy-numbers. If evaluated with interval arithmetic you can do very long calculations involving uncertain numbers very fast and with strong mathematical guarantees.
It would especially outperform the Monte-Carlo approach drastically.
I'm familiar with fuzzy numbers (e.g. see my https://filiph.net/fuzzy/ toy) but I didn't know there's arithmetic with fuzzy numbers. How is it done? Do you have a link?
There is a book by Hanss on it. It focuses on the sampling approach (he calls it "transformation method") though.
If you want to do arithmetic and not a black box approach you just have to realize that you can perform them on the alpha-cuts with ordinary interval arithmetic. Then you can evaluate arbitrary expressions involving fuzzy numbers, keeping the strengths and weaknesses of interval arithmetic.
The sampling based approach is very similar to Monte-Carlo, but you sample at certain well defined points.
Wow, this is fantastic! I did not know about squiggle language, and it's basically what I was trying to get to from my unsure calculator through my next project (https://filiph.github.io/napkin/). Squiggle looks and works much better.
Really cool! On iOS there's a noticeable delay when clicking the buttons and clicking the backspace button quickly zooms the page so it's very hard to use. Would love it in mobile friendly form!
Cool! Some random requests to consider: Could the range x~y be uniform instead of 2 std dev normal (95.4%ile)? Sometimes the range of quantities is known. 95%ile is probably fine as a default though.
Also, could a symbolic JS package be used instead of Monte-Carlo? This would improve speed and precision, especially for many variables (high dimensions).
Could the result be shown in a line plot instead of ASCII bar chart?
It's hard for me to imagine _dividing_ by -1~1 in a real-world scenario, but let's say we divide by 0~10, which also includes zero. For example, we are dividing the income between 0 to 10 shareholders (still forced, but ok).
Clearly, it's possible to have a division by zero here, so "0 sharehodlers would each get infinity". And in fact, if you try to compute 500 / 0, or even 500~1000 / 0, it will correctly show infinity.
But if you divide by a range that merely _includes_ zero, I don't think it should give you infinity. Ask yourself this: does 95% of results of 500 / 0~10 become infinity?
See also Guesstimate https://getguesstimate.com. Strengths include treating label and data as a unit, a space for examining the reasoning for a result, and the ability to replace an estimated distribution with sample data => you can build a model and then refine it over time. I'm amazed Excel and Google Sheets still haven't incorporated these things, years later.
I love this! As a tool for helping folks with a good base in arithmetic develop statistical intuition, I can't think offhand of what I've seen that's better.
i like it and i skimmed the post but i don't understand why the default example 100 / 4~6 has a median of 20? there is no way of knowing why the range is between 4 and 6
> Range is always a normal distribution, with the lower number being two standard deviations below the mean, and the upper number two standard deviations above. Nothing fancier is possible, in terms of input probability distributions.
There's an amazing scene in "This is Spinal Tap" where Nigel Tufnel had been brainstorming a scene where Stonehenge would be lowered from above onto the stage during their performance, and he does some back of the envelope calculations which he gives to the set designer. Unfortunately, he mixes the symbol for feet with the symbol for inches. Leading to the following:
In the grand HN tradition of being triggered by a word in the post and going off on a not-quite-but-basically-totally-tangential rant:
There’s (at least) three areas here that are footguns with these kinds of calculations:
1) 95% is usually a lot wider than people think - people take 95% as “I’m pretty sure it’s this,” whereas it’s really closer to “it’d be really surprising if it were not this” - by and large people keep their mental error bars too close.
2) probability is rarely truly uncorrelated - call this the “Mortgage Derivatives” maxim. In the family example, rent is very likely to be correlated with food costs - so, if rent is high, food costs are also likely to be high. This skews the distribution - modeling with an unweighted uniform distribution will lead to you being surprised at how improbable the actual outcome was.
3) In general normal distributions are rarer than people think - they tend to require some kind of constraining factor on the values to enforce. We see them a bunch in nature because there tends to be negative feedback loops all over the place, but once you leave the relatively tidy garden of Mother Nature for the chaos of human affairs, normal distributions get pretty abnormal.
I like this as a tool, and I like the implementation, I’ve just seen a lot of people pick up statistics for the first time and lose a finger.
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