Consider the following dialogue, between two forecasters aiming to predict the next presidential election:
Forecaster Alice: I believe that candidate X will win the election, with probability 36.5%.
Forecaster Bob: I believe that candidate X will win the election, with probability 30%-43%.
Forecaster Alice: What do you mean by giving such a wide range of values? Surely that just collapses to the mean of the range, which is my answer? If you were to offer me a bet that I could take either side of, surely the payoff that you would choose would be 36.5:63.5.
Forecaster Bob: I mean to say that I am uncertain about the probability. Forecasting is a hard and complicated job, and I must consider a lot of cases. I feel uncertain about this number that I give, so I give a confidence interval that I am reasonably certain that the true probability lies inside. Personally, I am disturbed that you gave such a specific answer. Do you think that you are so much better at forecasting than me, that you can give such a specific value? Have some humility.
Forecaster Alice: There is no such thing as a “true probability”, and uncertainty is represented by probability. If God were to look at the world, He would give a probability of either 0 or 1 to the outcome. It is already determined by a deterministic physics, we are just uncertain about which way that it is determined due to our uncertainty about the world. This uncertainty is demarcated in a probability. To give an uncertainty of an uncertainty implies that you are uncertain about what you, yourself, believe, which I do not think is accurate.
The goal of this post is to uncover what Forecaster Bob means, by giving a range of probabilities. Not exactly to determine whether Alice or Bob is right, but to Dissolve The Question, to determine exactly what generates the feeling of unease at giving a specific number in the mind of Forecaster Bob, even if he himself does not know.
I will meander through this post slightly, exploring some toy problems before giving the answer. For those who just want it quickly:
Forecaster Bob’s range of probabilities represents his uncertainty over what his probability might change to in the future, after seeing new evidence, or, equivalently, the width of his current probability distribution over the state of the world. He is not uncertain over what he currently believes, but over what he will believe in the future. He interprets Forecaster Alice only giving a number as a 0-width range[1], declaring that she is confident that she will not change her mind in the future, and that she is certain about the current state of the world.
If that made intuitive sense, feel free to skip the rest of the post, which is about justifying it. For everyone else, though:
Laplace’s Rule Of Succession
Imagine an urn containing a number of red or blue balls, which we draw out one at a time. Each draw can be red or blue, and balls are replaced after each draw. This urn has an “inherent frequency”, which is entirely physical, a function of how many balls of each colour are in the urn. Let us track our beliefs about the inherent frequency as we draw out balls of each colour.
Initially, we have seen no balls, and thus should start with a uniform prior over inherent frequencies. This looks like , on the domain , and is a probability density function. Then, say that we draw a single red ball. How our distribution over change? We answer this with Bayes’ rule,
Each individual frequency gets scaled by the probability that it assigned to the outcome we observed. Frequencies of red that were higher assigned more probability to the correct outcome, and so we believe that they are more likely, and this probability mass is taken from lower frequencies of red, which we now believe less. Our distribution is now skewed towards larger values of . In the extreme case, , as . We have totally ruled out there being no red balls in the urn.
As we pull out more balls, our distribution changes. The mean of the distribution will always be the observed frequency of balls, but it can be more or less tightly peaked, depending on how many balls we have observed. Consider two Bayesian reasoners, who both observed different sequences. Our first reasoner observed 6 red balls and 4 blue balls, while our second reasoner observed 60 red balls and 40 blue balls. These reasoners have different distributions over frequencies, with identical means but different variances.
If you ask each reasoner about the probability of a red ball being next, they will give the same answer, 60%. However, if each reasoner actually sees a red ball being next, they will update differently. The first reasoner, who has seen less evidence overall, and has a broader distribution, will update towards a higher frequency of red more strongly than the second reasoner, who has seen more evidence, and has a tighter distribution. We could say that the second reasoner is “more confident” in their prediction, as they change it less in response to new evidence.
Confidence implies an uncertainty, and in this case the uncertainty is over the value of , the inherent frequency. This setup is artificial, in that it contains an “irreducible” uncertainty, namely, that the value of does not actually determine the colour of the ball that is drawn. This is a frequency that lives in the world, in the proportions of balls in the urn, and so reasoners can have a probability (which lives in their head) over its different values. It is, however, very easy to confuse the quantity with , where is a specific value that can take, due to the fact that . This is the mistake that Forecaster Alice believes that Forecaster Bob is making, confusing his probability for some kind of physical “inherent frequency”, which has a definite value, but that he is uncertain over. Of course, a presidential election like this is a one-shot event, which cannot possibly have an inherent frequency. To end the discussion here, and conclude Forecaster Bob as simply committing the Mind Projection Fallacy, however, would be negligent. Forecaster Bob’s perspective can be rescued, and to see how, we shall turn to an example more representative of real life.
A One-Shot Urn Draw
Consider now an urn which contains a red ball or a blue ball, as well as a (hidden) dial, which can go from 0 to 1. If the dial is set higher than 0.5, a red ball will be drawn. If it is set less than 0.5, a blue ball. We seek to predict what single ball will be drawn out in the future, by maintaining a probability distribution over the position of the dial. To assist us with this, there are a number of signs and portents that appear to us. We observe that the urn has a spiral marking, which we know to be more likely to be true of urns with dials set to high numbers, but we also observe that the urn sits upon a square pedestal, which we intuitively associate with urns with dials set to low numbers.
How do we get these associations between signs and portents (which I will henceforth call evidence) and the position of the dial? We have some kind of model for how the dial is set, and we believe that the evidence that we see is correlated with that process. We might know that spirals put the urn attendant who set it up into a dial-pushing-right sort of mood, or squares into a dial-pushing-left mood. Thus, observing our evidence gives us information about where the dial is, and therefore the colour of the ball.
The dial is present to make this example similar to the previous one, where the urn has some inherent physical propensity to give out red or blue, although it is now deterministic. If we have a probability distribution over the position of the dial , that can still give us a probability of which ball we will draw out, given by .
We can again reconstruct the two reasoners observing the urn, one of which has seen 6 red portents and 4 blue portents, while the other has seen 60 red and 40 blue. These reasoners would still give the same , and would still update differently in response to seeing another portent. This is because the reasoners have different distributions over , with the width of each distribution determining the magnitude of update. To fully disclose each reasoner’s state of mind, the full distribution over the dial is needed, not just .
By eliminating the source of inherent randomness, we can see that just the value is still not all a reasoner knows. Given knowledge of the future evidence stream, they also have a guess for how much they will update in the future. The mean value of their future is of course just the present value, but the variance can vary greatly. The first reasoner would have a high variance of their future value of , while the second reasoner would have a low variance. The future value of their beliefs is uncertain to them in the present, so it is valid to have a probability distribution over it, with a certain variance.
The Election
With this groundwork established, we can look to the actual setup, of forecasting an election. The election results, in the future, are already determined by the state of the world that the forecasters live in in the present, and if they were omniscient, they would know with certainty the result. However, they are not omniscient, and have an uncertainty over the current state of the world, which then feeds forward into an uncertainty over the election results.
In the one-shot urn analogy, the position of the hidden dial is the current state of the world, and the colour of the ball drawn is the result of the election. The signs and portents that the urn-watchers receive are the observations that the forecasters make about the world today. Each forecaster has a probability distribution over the possible states that the world could be in, analogous to a probability distribution over the position of the dial, and to determine the probability that candidate X will be elected, they integrate that probability distribution over all the states that lead to candidate X being elected. Finally, we can determine what lead Forecaster Bob to give a range of probabilities. It was a range of what Forecaster Bob believes that his future beliefs will be, after seeing more evidence.
When Forecaster Bob feels uneasy about his number, it means that he anticipates changing it greatly in response to new evidence, which implies that his probability distribution over present states of the world is broad, that he is uncertain. Forecaster Alice, on the other hand, is not sharing the information about how variable her future probability might be, instead just giving her current probability. Forecaster Bob interpreted this as Forecaster Alice declaring that her range was in the next decimal place not given, from 36.45% to 36.55%. Having a range this small is declaring that you will not update very much at all on future evidence, that your probability distribution over present states of the world is sharply peaked, or relatively certain. This implication is why Forecaster Bob took offense to the perceived arrogance of seeing a single number of such precision.
Neither Alice nor Bob were incorrect in their use of probabilities, but simply communicating different information, with Alice communicating her scalar present probability, while Bob communicated the center and variance of his distribution over uncertain future probabilities.
Ranges of Probabilities: What Are They For?
Consider the following dialogue, between two forecasters aiming to predict the next presidential election:
The goal of this post is to uncover what Forecaster Bob means, by giving a range of probabilities. Not exactly to determine whether Alice or Bob is right, but to Dissolve The Question, to determine exactly what generates the feeling of unease at giving a specific number in the mind of Forecaster Bob, even if he himself does not know.
I will meander through this post slightly, exploring some toy problems before giving the answer. For those who just want it quickly:
Forecaster Bob’s range of probabilities represents his uncertainty over what his probability might change to in the future, after seeing new evidence, or, equivalently, the width of his current probability distribution over the state of the world. He is not uncertain over what he currently believes, but over what he will believe in the future. He interprets Forecaster Alice only giving a number as a 0-width range[1], declaring that she is confident that she will not change her mind in the future, and that she is certain about the current state of the world.
If that made intuitive sense, feel free to skip the rest of the post, which is about justifying it. For everyone else, though:
Laplace’s Rule Of Succession
Imagine an urn containing a number of red or blue balls, which we draw out one at a time. Each draw can be red or blue, and balls are replaced after each draw. This urn has an “inherent frequency”, which is entirely physical, a function of how many balls of each colour are in the urn. Let us track our beliefs about the inherent frequency as we draw out balls of each colour.
Initially, we have seen no balls, and thus should start with a uniform prior over inherent frequencies. This looks like
Each individual frequency gets scaled by the probability that it assigned to the outcome we observed. Frequencies of red that were higher assigned more probability to the correct outcome, and so we believe that they are more likely, and this probability mass is taken from lower frequencies of red, which we now believe less. Our distribution is now skewed towards larger values of
As we pull out more balls, our distribution changes. The mean of the distribution will always be the observed frequency of balls, but it can be more or less tightly peaked, depending on how many balls we have observed. Consider two Bayesian reasoners, who both observed different sequences. Our first reasoner observed 6 red balls and 4 blue balls, while our second reasoner observed 60 red balls and 40 blue balls. These reasoners have different distributions over frequencies, with identical means but different variances.
If you ask each reasoner about the probability of a red ball being next, they will give the same answer, 60%. However, if each reasoner actually sees a red ball being next, they will update differently. The first reasoner, who has seen less evidence overall, and has a broader distribution, will update towards a higher frequency of red more strongly than the second reasoner, who has seen more evidence, and has a tighter distribution. We could say that the second reasoner is “more confident” in their prediction, as they change it less in response to new evidence.
Confidence implies an uncertainty, and in this case the uncertainty is over the value of
A One-Shot Urn Draw
Consider now an urn which contains a red ball or a blue ball, as well as a (hidden) dial, which can go from 0 to 1. If the dial is set higher than 0.5, a red ball will be drawn. If it is set less than 0.5, a blue ball. We seek to predict what single ball will be drawn out in the future, by maintaining a probability distribution over the position of the dial. To assist us with this, there are a number of signs and portents that appear to us. We observe that the urn has a spiral marking, which we know to be more likely to be true of urns with dials set to high numbers, but we also observe that the urn sits upon a square pedestal, which we intuitively associate with urns with dials set to low numbers.
How do we get these associations between signs and portents (which I will henceforth call evidence) and the position of the dial? We have some kind of model for how the dial is set, and we believe that the evidence that we see is correlated with that process. We might know that spirals put the urn attendant who set it up into a dial-pushing-right sort of mood, or squares into a dial-pushing-left mood. Thus, observing our evidence gives us information about where the dial is, and therefore the colour of the ball.
The dial is present to make this example similar to the previous one, where the urn has some inherent physical propensity to give out red or blue, although it is now deterministic. If we have a probability distribution over the position of the dial
We can again reconstruct the two reasoners observing the urn, one of which has seen 6 red portents and 4 blue portents, while the other has seen 60 red and 40 blue. These reasoners would still give the same
By eliminating the source of inherent randomness, we can see that just the
The Election
With this groundwork established, we can look to the actual setup, of forecasting an election. The election results, in the future, are already determined by the state of the world that the forecasters live in in the present, and if they were omniscient, they would know with certainty the result. However, they are not omniscient, and have an uncertainty over the current state of the world, which then feeds forward into an uncertainty over the election results.
In the one-shot urn analogy, the position of the hidden dial is the current state of the world, and the colour of the ball drawn is the result of the election. The signs and portents that the urn-watchers receive are the observations that the forecasters make about the world today. Each forecaster has a probability distribution over the possible states that the world could be in, analogous to a probability distribution over the position of the dial, and to determine the probability that candidate X will be elected, they integrate that probability distribution over all the states that lead to candidate X being elected. Finally, we can determine what lead Forecaster Bob to give a range of probabilities. It was a range of what Forecaster Bob believes that his future beliefs will be, after seeing more evidence.
When Forecaster Bob feels uneasy about his number, it means that he anticipates changing it greatly in response to new evidence, which implies that his probability distribution over present states of the world is broad, that he is uncertain. Forecaster Alice, on the other hand, is not sharing the information about how variable her future probability might be, instead just giving her current probability. Forecaster Bob interpreted this as Forecaster Alice declaring that her range was in the next decimal place not given, from 36.45% to 36.55%. Having a range this small is declaring that you will not update very much at all on future evidence, that your probability distribution over present states of the world is sharply peaked, or relatively certain. This implication is why Forecaster Bob took offense to the perceived arrogance of seeing a single number of such precision.
Neither Alice nor Bob were incorrect in their use of probabilities, but simply communicating different information, with Alice communicating her scalar present probability, while Bob communicated the center and variance of his distribution over uncertain future probabilities.
More accurately, a range over the next decimal place, from 36.45% to 36.55%, which is small.
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