I’m contemplating a discussion post on this topic, but first I’ll float it here, since there’s a high chance that I’m just being really stupid.
I’m abysmally unsuccessful at using anything like Bayesian reasoning in real life.
I don’t think it’s because I’m doing anything fundamentally wrong. Maybe what I’m doing wrong is attempting to think of these things in a Bayesian way in the first place.
Let’s use a concrete example. I bought a house. My prior probability that any given household appliance or fixture will break and/or need maintenance in a given month is on the order of 5%, obviously with some variability depending on what appliance we’re talking about. This prior is an off-the-cuff intuitive figure based on decades of living in houses.
Within a month of buying this house, things immediately start breaking. The dishwasher breaks. Then the garbage disposal. The sump pump fails completely. The humidifier needs repair. The air conditioner unit needs to be entirely replaced. The siding needs to be repainted. A section of fence needs to be replaced. The sprinklers don’t work. This is all within roughly the first four months.
So, my prior was garbage, but the real issue for me is that Bayesian reasoning didn’t really help me. The dishwasher breaking didn’t cause me to shift my Background Probabilistic Breakage Rate much at all. One thing breaking within the first month is allowed for by my prior model. Then the second thing breaks—okay, maybe I need to adjust my BPBR a a bit. Still, there’s little reason to expect that several more important things will break in short order. But that’s exactly what happened.
There is a causal story that explains everything (apparently) breaking at basically the same time, which is that the previous owners were not taking good care of the house, and various things were already subtly broken and limping along at passable functionality for a long time. The problem is that this causal story only becomes promoted to “hypothesis with significant probability mass” after two or three consecutive major appliance disasters.
What is annoying about all this is that my wife doesn’t attempt to use any kind of probabilistic reasoning, and she is basically right all the time. I was saying things like, “I really doubt the garbage disposal is really broken, we just had two other major things replaced, what are the odds that another thing would break so quickly?” and she would reply along the lines of, “I’m pretty sure it’s actually broken, and I can’t fathom why you keep talking about odds when your odds-based assessments are always wrong,” and I’m at the point of agreeing with her. Not to mention that she was the one who suggested the “prior owners didn’t maintain the house” hypothesis, while I was still grimly clinging to my initial model, increasingly bewildered by each new disaster.
I am probably a poster child for “doing probabilistic thinking wrong” in some obvious way that I am blind to. Please help me figure out how and where. I have my own thoughts, but I will wait for others to respond so as to avoid anchoring.
I think you were basically doing okay, it’s just that as soon as you formulated your initial hypothesis you should have actively sought out a way to disprove it. How hard can I lean on my fence? Is scratching lazily sufficient to remove paint on the siding? Do I dare to wash the floor under the bookshelf?… After all, if you suddenly received lots of evidence to the contrary, you would a) update fast, and b) earn husband points.
In essence, you should always ask yourself, is this still the relevant question?
As a bounded agent, you have to be aware that it’s physically impossible to consider all the hypotheses. When you encounter new evidence, you might think of a new hypothesis to promote that you hadn’t thought of before—in fact, this is an unavoidable part of being a good bounded agent. So don’t worry about coming up with the One True Prior ahead of time and then updating it—instead, try to plan for the most likely outcomes, but leave a “something else” category and be ready to change your mind.
And given that we’re biased, when we make plans we’re probably going to get some probabilities wrong—in this case, future events contain information about how one was biased. Try to learn about your own biases, which often means being more influenced by evidence than an unbiased agent.
If you still want to try reasoning probabilistically, I’d look into Tetlock’s good judgment project and start planning how to practice my probability estimation. Oh, and check out the calibration game.
I was saying things like, “I really doubt the garbage disposal is really broken, we just had two other major things replaced, what are the odds that another thing would break so quickly?” [...] I am probably a poster child for “doing probabilistic thinking wrong” in some obvious way that I am blind to.
You are indeed doing it very wrong. As far as proablisitic reasoning goes the fact that one item broke doesn’t reduce the chances that a second item breaks at all.
Yeah, okay, I worded that stupidly. It’s more like this:
“This 20-sided-die just came up 20 twice in a row. The odds of three consecutive rolls of 20 is 0.0125%. I acknowledge that this next roll has a 1⁄20 chance of coming up 20, assuming the die is fair. However, if this next roll comes up 20, we are witnessing an extremely improbable sequence, so improbable that I have to start considering that the die is loaded.”
However, if this next roll comes up 20, we are witnessing an extremely improbable sequence, so improbable that I have to start considering that the die is loaded.”
The equivalent of “considering that the die is loaded” in your example is “the previous owners did a bad job of maintaining the house”. It’s indeed makes sense to come to that conclusion. That’s also basically what your wife did.
Apart from that the difference between sequences picked by humans to look random and real random data is that real random data more frequently contains such improbable sequences.
I mean, regardless of what were the previous two rolls—let’s call them “X” and “Y”—if the next roll comes up 20, we are witnessing a sequence “X, Y, 20”, which has a probability 0.0125%. That’s true even when “X” and “Y” are different than 20.
You could make the sequence even more improbable by saying “if this next roll comes up, we are witnessing an extremely improbably sequence—we are living in a universe whose conditions allow creation of matter, we happen to be on a planet where life exists, dinousaurs were killed by a comet, I decided to roll the 20-sided-die three times, the first two rolls were 20… and now the third roll is also 20? Well this all just seems very very unlikely.”
Or you could decide that the past is fixed, if you happen to be in some branch of the universe you are already there, and you are only going to estimate the probability of future events.
Even better, what ChristianKl said. A better model would be that depending on the existing state of the house there is a probability P saying how frequently things will break. At the beginning there is some prior distribution of P, but when things start breaking too fast, you should update that P is probably greater than you originally thought… and now you should expect things to break faster than you expected originally.
regardless of what were the previous two rolls—let’s call them “X” and “Y”—if the next roll comes up 20, we are witnessing a sequence “X, Y, 20”, which has a probability 0.0125%. That’s true even when “X” and “Y” are different than 20.
Yes, all sequences X,Y,Z are equally (im)probable if the d20 is a fair one. But some sequences—in particular those with X=Y=Z, and in more-particular those with X=Y=Z=1 or X=Y=Z=20, are more likely if the die is unfair because they’re relatively easy and/or relatively useful/amusing for a die-fixer to induce.
As you consider longer and longer sequences 20,20,20,… their probability conditional on a fair d20 goes down rapidly, whereas their probability conditional on a dishonest d20 goes down much less rapidly because there’s some nonzero chance that someone’s made a d20 that almost always rolls 20s.
separate advice: look around at things and check if anything else is about to break and can be saved from expensive replacement via the process of repairs.
I am probably a poster child for “doing probabilistic thinking wrong” in some obvious way that I am blind to.
One possible mistake is assuming that problems will be independent and spread out evenly over time. That’s an extreme assumption. In real life there are always more reasons for problems to cluster than to anti-cluster (so to speak), so it doesn’t balance out at all. Also, problems will do more harm when clustered, because your ability to cope is reduced. So it makes sense to prepare for clustered problems. When two things go wrong, get ready for the third. That’s very obvious in software engineering, if you find ten bugs, chances are you haven’t found them all. But it’s true in real life too.
The more general problem is that you just seem to have less life experience than your wife. To fix that, go out and get experience. Fix stuff, haggle, make arrangements… It’ll improve your life in other ways as well.
“I really doubt the garbage disposal is really broken, we just had two other major things replaced, what are the odds that another thing would break so quickly?”
You have two hypotheses: the appliances breaking are not connected (independent); and the appliance breaking are connected (dependent).
In the first case you are saying the equivalent of “I tossed the coin twice and it came up heads both times, it’s really unlikely it will come up heads the third time as well” which should be obviously wrong.
In the second case you should discard your model of independence alongside with your original prior and consider that the breakages are connected.
I think the moral of the story is that life is complicated and simple models are often too simple to be useful. You should discard them faster when they show signs of not working.
And, of course, if you are wondering whether your garbage disposal is really broken, you should go look at your garbage disposal unit and not engage in pondering theoretical considerations.
See my response to ChristianKl below for my clarification on my reasoning about “consecutive coin flips” which could still be wrong but is hopefully less wrong than my original wording.
I agree that I should have discarded my model more quickly, but I don’t quite see how to generalize that observation. Sometimes the alternative hypothesis (e.g. the breakages are connected) is not apparent or obvious without more data—and the process of collecting data really just means continuing to make bad predictions as you go through life until something clicks and you notice the underlying structure.
My wife seems to think that making explicit model-based predictions in the first place is the problem. I have a lot of respect for System 1 and am sympathetic to this view. But System 2 really shouldn’t actively lead me astray.
Yes, and note that this part—“that I have to start considering that the die is loaded”—is key.
but I don’t quite see how to generalize that observation
Um, directly? All models which you are considering are much simpler than the real world. The relevant maxim is “All models are wrong, but some are useful”.
I think you got caught in the trap of “but I can’t change my prior because priors are not supposed to be changed”. That’s not exactly true. You can and (given sufficient evidence) should be willing to discard your entire model and the prior with it. Priors only make sense within a specified set of hypotheses. If your set of hypotheses changes, the old prior goes out of the window.
The naive Bayes approach sweeps a lot of complexity under the rug (e.g. hypotheses selection) which will bite you in the ass given the slightest opportunity.
Sometimes the alternative hypothesis (e.g. the breakages are connected) is not apparent or obvious
Yeah, well, welcome to the real world :-/
My wife seems to think that making explicit model-based predictions in the first place is the problem.
She is correct if your models are wrong. Getting right models is hard and you should not assume that the first model you came up with is going to be sufficiently correct to be useful.
System 2 really shouldn’t actively lead me astray.
I see absolutely no basis for this belief. To misquote someone from memory: “Logic is just a way of making errors with confidence” :-P
I really doubt the garbage disposal is really broken, we just had two other major things replaced, what are the odds that another thing would break so quickly?
If they’re independent, the odds are exactly the same as if two other major things had not been recently replaced. If they’re dependent, the odds are higher. For this decision, you have a lot more evidence about the specific, so your base rate (unless it’s incredibly small) doesn’t matter, and the specific evidence of brokenness overwhelms it.
A better application is budgeting for next month. Compare how much you’re planning to set aside for repairs with how much your wife is. See who’s right. Update. Repeat.
I can’t help but notice, in an slightly off-topic fugue, that the dishwasher, the garbage disposal, and probably the sump pump share a drainage system. You may wish to consider the possibility that these are not independent breakages, and that until you fix the underlying problem, you should expect further breakages (i.e., check you drains).
Also, the siding needing to be repainted and a section of fence needing to be replaced doesn’t really sound like “things breaking” (I could be wrong). Could you have been ignoring some important information right from the start?
My prior probability that any given household appliance or fixture will break and/or need maintenance in a given month is on the order of 5%, obviously with some variability depending on what appliance we’re talking about.
There’s your problem right there. Note, that this prior effectively assigns zero probability to the “prior owner didn’t maintain the house” hypothesis.
What you should have done is assign some (non-zero) probability to that hypothesis, then when something breaks, one would updates towards the “poor maintenance” hypothesis.
I’m contemplating a discussion post on this topic, but first I’ll float it here, since there’s a high chance that I’m just being really stupid.
I’m abysmally unsuccessful at using anything like Bayesian reasoning in real life.
I don’t think it’s because I’m doing anything fundamentally wrong. Maybe what I’m doing wrong is attempting to think of these things in a Bayesian way in the first place.
Let’s use a concrete example. I bought a house. My prior probability that any given household appliance or fixture will break and/or need maintenance in a given month is on the order of 5%, obviously with some variability depending on what appliance we’re talking about. This prior is an off-the-cuff intuitive figure based on decades of living in houses.
Within a month of buying this house, things immediately start breaking. The dishwasher breaks. Then the garbage disposal. The sump pump fails completely. The humidifier needs repair. The air conditioner unit needs to be entirely replaced. The siding needs to be repainted. A section of fence needs to be replaced. The sprinklers don’t work. This is all within roughly the first four months.
So, my prior was garbage, but the real issue for me is that Bayesian reasoning didn’t really help me. The dishwasher breaking didn’t cause me to shift my Background Probabilistic Breakage Rate much at all. One thing breaking within the first month is allowed for by my prior model. Then the second thing breaks—okay, maybe I need to adjust my BPBR a a bit. Still, there’s little reason to expect that several more important things will break in short order. But that’s exactly what happened.
There is a causal story that explains everything (apparently) breaking at basically the same time, which is that the previous owners were not taking good care of the house, and various things were already subtly broken and limping along at passable functionality for a long time. The problem is that this causal story only becomes promoted to “hypothesis with significant probability mass” after two or three consecutive major appliance disasters.
What is annoying about all this is that my wife doesn’t attempt to use any kind of probabilistic reasoning, and she is basically right all the time. I was saying things like, “I really doubt the garbage disposal is really broken, we just had two other major things replaced, what are the odds that another thing would break so quickly?” and she would reply along the lines of, “I’m pretty sure it’s actually broken, and I can’t fathom why you keep talking about odds when your odds-based assessments are always wrong,” and I’m at the point of agreeing with her. Not to mention that she was the one who suggested the “prior owners didn’t maintain the house” hypothesis, while I was still grimly clinging to my initial model, increasingly bewildered by each new disaster.
I am probably a poster child for “doing probabilistic thinking wrong” in some obvious way that I am blind to. Please help me figure out how and where. I have my own thoughts, but I will wait for others to respond so as to avoid anchoring.
I think you were basically doing okay, it’s just that as soon as you formulated your initial hypothesis you should have actively sought out a way to disprove it. How hard can I lean on my fence? Is scratching lazily sufficient to remove paint on the siding? Do I dare to wash the floor under the bookshelf?… After all, if you suddenly received lots of evidence to the contrary, you would a) update fast, and b) earn husband points.
In essence, you should always ask yourself, is this still the relevant question?
Some random thoughts:
As a bounded agent, you have to be aware that it’s physically impossible to consider all the hypotheses. When you encounter new evidence, you might think of a new hypothesis to promote that you hadn’t thought of before—in fact, this is an unavoidable part of being a good bounded agent. So don’t worry about coming up with the One True Prior ahead of time and then updating it—instead, try to plan for the most likely outcomes, but leave a “something else” category and be ready to change your mind.
And given that we’re biased, when we make plans we’re probably going to get some probabilities wrong—in this case, future events contain information about how one was biased. Try to learn about your own biases, which often means being more influenced by evidence than an unbiased agent.
If you still want to try reasoning probabilistically, I’d look into Tetlock’s good judgment project and start planning how to practice my probability estimation. Oh, and check out the calibration game.
You are indeed doing it very wrong. As far as proablisitic reasoning goes the fact that one item broke doesn’t reduce the chances that a second item breaks at all.
Yeah, okay, I worded that stupidly. It’s more like this:
“This 20-sided-die just came up 20 twice in a row. The odds of three consecutive rolls of 20 is 0.0125%. I acknowledge that this next roll has a 1⁄20 chance of coming up 20, assuming the die is fair. However, if this next roll comes up 20, we are witnessing an extremely improbable sequence, so improbable that I have to start considering that the die is loaded.”
The equivalent of “considering that the die is loaded” in your example is “the previous owners did a bad job of maintaining the house”. It’s indeed makes sense to come to that conclusion. That’s also basically what your wife did.
Apart from that the difference between sequences picked by humans to look random and real random data is that real random data more frequently contains such improbable sequences.
The “however” part seems irrelevant.
I mean, regardless of what were the previous two rolls—let’s call them “X” and “Y”—if the next roll comes up 20, we are witnessing a sequence “X, Y, 20”, which has a probability 0.0125%. That’s true even when “X” and “Y” are different than 20.
You could make the sequence even more improbable by saying “if this next roll comes up, we are witnessing an extremely improbably sequence—we are living in a universe whose conditions allow creation of matter, we happen to be on a planet where life exists, dinousaurs were killed by a comet, I decided to roll the 20-sided-die three times, the first two rolls were 20… and now the third roll is also 20? Well this all just seems very very unlikely.”
Or you could decide that the past is fixed, if you happen to be in some branch of the universe you are already there, and you are only going to estimate the probability of future events.
Even better, what ChristianKl said. A better model would be that depending on the existing state of the house there is a probability P saying how frequently things will break. At the beginning there is some prior distribution of P, but when things start breaking too fast, you should update that P is probably greater than you originally thought… and now you should expect things to break faster than you expected originally.
Yes, all sequences X,Y,Z are equally (im)probable if the d20 is a fair one. But some sequences—in particular those with X=Y=Z, and in more-particular those with X=Y=Z=1 or X=Y=Z=20, are more likely if the die is unfair because they’re relatively easy and/or relatively useful/amusing for a die-fixer to induce.
As you consider longer and longer sequences 20,20,20,… their probability conditional on a fair d20 goes down rapidly, whereas their probability conditional on a dishonest d20 goes down much less rapidly because there’s some nonzero chance that someone’s made a d20 that almost always rolls 20s.
separate advice: look around at things and check if anything else is about to break and can be saved from expensive replacement via the process of repairs.
One possible mistake is assuming that problems will be independent and spread out evenly over time. That’s an extreme assumption. In real life there are always more reasons for problems to cluster than to anti-cluster (so to speak), so it doesn’t balance out at all. Also, problems will do more harm when clustered, because your ability to cope is reduced. So it makes sense to prepare for clustered problems. When two things go wrong, get ready for the third. That’s very obvious in software engineering, if you find ten bugs, chances are you haven’t found them all. But it’s true in real life too.
The more general problem is that you just seem to have less life experience than your wife. To fix that, go out and get experience. Fix stuff, haggle, make arrangements… It’ll improve your life in other ways as well.
You have two hypotheses: the appliances breaking are not connected (independent); and the appliance breaking are connected (dependent).
In the first case you are saying the equivalent of “I tossed the coin twice and it came up heads both times, it’s really unlikely it will come up heads the third time as well” which should be obviously wrong.
In the second case you should discard your model of independence alongside with your original prior and consider that the breakages are connected.
I think the moral of the story is that life is complicated and simple models are often too simple to be useful. You should discard them faster when they show signs of not working.
And, of course, if you are wondering whether your garbage disposal is really broken, you should go look at your garbage disposal unit and not engage in pondering theoretical considerations.
See my response to ChristianKl below for my clarification on my reasoning about “consecutive coin flips” which could still be wrong but is hopefully less wrong than my original wording.
I agree that I should have discarded my model more quickly, but I don’t quite see how to generalize that observation. Sometimes the alternative hypothesis (e.g. the breakages are connected) is not apparent or obvious without more data—and the process of collecting data really just means continuing to make bad predictions as you go through life until something clicks and you notice the underlying structure.
My wife seems to think that making explicit model-based predictions in the first place is the problem. I have a lot of respect for System 1 and am sympathetic to this view. But System 2 really shouldn’t actively lead me astray.
Yes, and note that this part—“that I have to start considering that the die is loaded”—is key.
Um, directly? All models which you are considering are much simpler than the real world. The relevant maxim is “All models are wrong, but some are useful”.
I think you got caught in the trap of “but I can’t change my prior because priors are not supposed to be changed”. That’s not exactly true. You can and (given sufficient evidence) should be willing to discard your entire model and the prior with it. Priors only make sense within a specified set of hypotheses. If your set of hypotheses changes, the old prior goes out of the window.
The naive Bayes approach sweeps a lot of complexity under the rug (e.g. hypotheses selection) which will bite you in the ass given the slightest opportunity.
Yeah, well, welcome to the real world :-/
She is correct if your models are wrong. Getting right models is hard and you should not assume that the first model you came up with is going to be sufficiently correct to be useful.
I see absolutely no basis for this belief. To misquote someone from memory: “Logic is just a way of making errors with confidence” :-P
If they’re independent, the odds are exactly the same as if two other major things had not been recently replaced. If they’re dependent, the odds are higher. For this decision, you have a lot more evidence about the specific, so your base rate (unless it’s incredibly small) doesn’t matter, and the specific evidence of brokenness overwhelms it.
A better application is budgeting for next month. Compare how much you’re planning to set aside for repairs with how much your wife is. See who’s right. Update. Repeat.
I can’t help but notice, in an slightly off-topic fugue, that the dishwasher, the garbage disposal, and probably the sump pump share a drainage system. You may wish to consider the possibility that these are not independent breakages, and that until you fix the underlying problem, you should expect further breakages (i.e., check you drains).
Also, the siding needing to be repainted and a section of fence needing to be replaced doesn’t really sound like “things breaking” (I could be wrong). Could you have been ignoring some important information right from the start?
There’s your problem right there. Note, that this prior effectively assigns zero probability to the “prior owner didn’t maintain the house” hypothesis.
What you should have done is assign some (non-zero) probability to that hypothesis, then when something breaks, one would updates towards the “poor maintenance” hypothesis.