The history of lighting technology

Behold, one of my favorite tables of all time:

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(Source.)

There’s so much to absorb here. Let’s look at just the “Light Price in Terms of Labor” column. At 500,000 BC, our starting point, we have this handsome guy:

Peking Man

The Peking man was a Homo erectus found in a cave with evidence of tool use and basic fire technology.  At this point, it would have taken him about 58 hours of work for every 1000 hours of light. Lighting a fire by hand or even with basic tools is hard, as seen here:

Nothing much changes for hundreds of thousands of years, until people begin using basic candles and oil lamps in the 1800s. After that, things slowly begin to accelerate, with gas lighting, incandescent lamps, and eventually fluorescent bulbs and LEDs…

Lights

Notice that this is a logarithmic plot! So a straight line corresponds to an exponential decrease in the amount of labor required to produce light. By the end we have less than 1 second to produce 1000 hours of light. And this doesn’t even include LED technologies!

Here’s a more detailed timeline of milestones in lighting technology up to the 1980s:

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And finally, a comparison of the efficiency of different lighting technologies over time.

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Kant’s attempt to save metaphysics and causality from Hume

TL;DR

  • Hume sort of wrecked metaphysics. This inspired Kant to try and save it.
  • Hume thought that terms were only meaningful insofar as they were derived from experience.
  • We never actually experience necessary connections between events, we just see correlations. So Hume thought that the idea of causality as necessary connection is empty and confused, and that all our idea of causality really amounts to is correlation.
  • Kant didn’t like this. He wanted to PROTECT causality. But how??
  • Kant said that metaphysical knowledge was both a priori and substantive, and justified this by describing these things called pure intuitions and pure concepts.
  • Intuitions are representations of things (like sense perceptions). Pure intuitions are the necessary preconditions for us to represent things at all.
  • Concepts are classifications of representations (like “red”). Pure concepts are the necessary preconditions underlying all classifications of representations.
  • There are two pure intuitions (space and time) and twelve pure concepts (one of which is causality).
  • We get substantive a priori knowledge by referring to pure intuitions (mathematics) or pure concepts (laws of nature, metaphysics).
  • Yay! We saved metaphysics!

 

(Okay, now on to the actual essay. This was not originally written for this blog, which is why it’s significantly more formal than my usual fare.)

 

***

 

David Hume’s Enquiry Into Human Understanding stands out as a profound and original challenge to the validity of metaphysical knowledge. Part of the historical legacy of this work is its effect on Kant, who describes Hume as being responsible for [interrupting] my dogmatic slumber and [giving] my investigations in the field of speculative philosophy a completely different direction.” Despite the great inspiration that Kant took from Hume’s writing, their thinking on many matters is diametrically opposed. A prime example of this is their views on causality.

Hume’s take on causation is famously unintuitive. He gives a deflationary account of the concept, arguing that the traditional conception lacks a solid epistemic foundation and must be replaced by mere correlation. To understand this conclusion, we need to back up and consider the goal and methodology of the Enquiry.

He starts with an appeal to the importance of careful and accurate reasoning in all areas of human life, and especially in philosophy. In a beautiful bit of prose, he warns against the danger of being overwhelmed by popular superstition and religious prejudice when casting one’s mind towards the especially difficult and abstruse questions of metaphysics.

But this obscurity in the profound and abstract philosophy is objected to, not only as painful and fatiguing, but as the inevitable source of uncertainty and error. Here indeed lies the most just and most plausible objection against a considerable part of metaphysics, that they are not properly a science, but arise either from the fruitless efforts of human vanity, which would penetrate into subjects utterly inaccessible to the understanding, or from the craft of popular superstitions, which, being unable to defend themselves on fair ground, raise these entangling brambles to cover and protect their weakness. Chased from the open country, these robbers fly into the forest, and lie in wait to break in upon every unguarded avenue of the mind, and overwhelm it with religious fears and prejudices. The stoutest antagonist, if he remit his watch a moment, is oppressed. And many, through cowardice and folly, open the gates to the enemies, and willingly receive them with reverence and submission, as their legal sovereigns.

In less poetic terms, Hume’s worry about metaphysics is that its difficulty and abstruseness makes its practitioners vulnerable to flawed reasoning. Even worse, the difficulty serves to make the subject all the more tempting for “each adventurous genius[, who] will still leap at the arduous prize and find himself stimulated, rather than discouraged by the failures of his predecessors, while he hopes that the glory of achieving so hard an adventure is reserved for him alone.”

Thus, says Hume, the only solution is “to free learning at once from these abstruse questions [by inquiring] seriously into the nature of human understanding and [showing], from an exact analysis of its powers and capacity, that it is by no means fitted for such remote and abstruse questions.”

Here we get the first major divergence between Kant and Hume. Kant doesn’t share Hume’s eagerness to banish metaphysics. His Prolegomena To Any Future Metaphysics and Critique of Pure Reason are attempts to find it a safe haven from Hume’s attacks. However, while Kant might not be similarly constituted to Hume in this way, he does take Hume’s methodology very seriously. He states in the preface to the Prolegomena that “since the origin of metaphysics as far as history reaches, nothing has ever happened which could have been more decisive to its fate than the attack made upon it by David Hume.” Many of the principles which Hume derives, Kant agrees with wholeheartedly, making the task of shielding metaphysics even harder for him.

With that understanding of Hume’s methodology in mind, let’s look at how he argues for his view of causality. We’ll start with a distinction that is central to Hume’s philosophy: that between ideas and impressions. The difference between the memory of a sensation and the sensation itself is a good example. While the memory may mimic or copy the sensation, it can never reach its full force and vivacity. In general, Hume suggests that our experiences fall into two distinct categories, separated by a qualitative gap in liveliness. The more lively category he calls impressions, which includes sensory perceptions like the smell of a rose or the taste of wine, as well as internal experiences like the feeling of love or anger. The less lively category he refers to as thoughts or ideas. These include memories of impressions as well as imagined scenes, concepts, and abstract thoughts. 

With this distinction in hand, Hume proposes his first limit on the human mind. He claims that no matter how creative or original you are, all of your thoughts are the product of “compounding, transposing, augmenting, or diminishing the materials afforded us by the senses and experiences.” This is the copy principle: all ideas are copies of impressions, or compositions of simpler ideas that are in turn copies of impressions.

Hume turns this observation of the nature of our mind into a powerful criterion of meaning. “When we entertain any suspicion that a philosophical term is employed without any meaning or idea (as is but too frequent), we need but enquire, From what impression is that supposed idea derived? And if it be impossible to assign any, this will serve to confirm our suspicion.

This criterion turns out to be just the tool Hume needs in order to establish his conclusion. He examines the traditional conception of causation as a necessary connection between events, searches for the impressions that might correspond to this idea, and, failing to find anything satisfactory, declares that “we have no idea of connection or power at all and that these words are absolutely without any meaning when employed in either philosophical reasonings or common life.” His primary argument here is that all of our observations are of mere correlation, and that we can never actually observe a necessary connection.

Interestingly, at this point he refrains from recommending that we throw out the term causation. Instead he proposes a redefinition of the term, suggesting a more subtle interpretation of his criterion of meaning. Rather than eliminating the concept altogether upon discovering it to have no satisfactory basis in experience, he reconceives it in terms of the impressions from which it is actually formed. In particular, he argues that our idea of causation is really based on “the connection which we feel in the mind, this customary transition of the imagination from one object to its usual attendant.”

Here Hume is saying that humans have a rationally unjustifiable habit of thought where, when we repeatedly observe one type of event followed by another, we begin to call the first a cause and the second its effect, and we expect that future instances of the cause will be followed by future instances of the effect. Causation, then, is just this constant conjunction between events, and our mind’s habit of projecting the conjunction into the future. We can summarize all of this in a few lines:

Hume’s denial of the traditional concept of causation

  1. Ideas are always either copies of impressions or composites of simpler ideas that are copies of impressions.
  2. The traditional conception of causation is neither of these.
  3. So we have no idea of the traditional conception of causation.

Hume’s reconceptualization of causation

  1. An idea is the idea of the impression that it is a copy of.
  2. The idea of causation is copied from the impression of constant conjunction.
  3. So the idea of causation is just the idea of constant conjunction.

There we have Hume’s line of reasoning, which provoked Kant to examine the foundations of metaphysics anew. Kant wanted to resist Hume’s dismissal of the traditional conception of causation, while accepting that our sense perceptions reveal no necessary connections to us. Thus his strategy was to deny the Copy Principle and give an account of how we can have substantive knowledge that is not ultimately traceable to impressions. He does this by introducing the analytic/synthetic distinction and the notion of a priori synthetic knowledge.

Kant’s original definition of analytic judgments is that they “express nothing in the predicate but what has already been actually thought in the concept of the subject.” This suggests that the truth value of an analytic judgment is determined by purely the meanings of the concepts in use. A standard example of this is “All bachelors are unmarried.” The truth of this statement follows immediately just by understanding what it means, as the concept of bachelor already contains the predicate unmarried.  Synthetic judgments, on the other hand, are not fixed in truth value by merely the meanings of the concepts in use. These judgments amplify our knowledge and bring us to genuinely new conclusions about our concepts. An example: “The President is ornery.” This certainly doesn’t follow by definition; you’d have to go out and watch the news to realize its truth.

We can now put the challenge to metaphysics slightly differently. Metaphysics purports to be discovering truths that are both necessary (and therefore a priori) as well as substantive (adding to our concepts and thus synthetic). But this category of synthetic a priori judgments seems a bit mysterious. Evidently, the truth values of such judgments can be determined without referring to experience, but can’t be determined by merely the meanings of the relevant concepts. So apparently something further is required besides the meanings of concepts in order to make a synthetic a priori judgment. What is this thing?

Kant’s answer is that the further requirement is pure intuition and pure concepts. These terms need explanation.

Pure Intuitions

For Kant, an intuition is a direct, immediate representation of an object. An obvious example of this is sense perception; looking at a cup gives you a direct and immediate representation of an object, namely, the cup. But pure intuitions must be independent of experience, or else judgments based on them would not be a priori. In other words, the only type of intuition that could possibly be a priori is one that is present in all possible perceptions, so that its existence is not contingent upon what perceptions are being had. Kant claims that this is only possible if pure intuitions represent the necessary preconditions for the possibility of perception.

What are these necessary preconditions? Kant famously claimed that the only two are space and time. This implies that all of our perceptions have spatiotemporal features, and indeed that perception is only possible in virtue of the existence of space and time. It also implies, according to Kant, that space and time don’t exist outside of our minds!  Consider that pure intuitions exist equally in all possible perceptions and thus are independent of the actual properties of external objects. This independence suggests that rather than being objective features of the external world, space and time are structural features of our minds that frame all our experiences.

This is why Kant’s philosophy is a species of idealism. Space and time get turned into features of the mind, and correspondingly appearances in space and time become internal as well. Kant forcefully argues that this view does not make space and time into illusions, saying that without his doctrine “it would be absolutely impossible to determine whether the intuitions of space and time, which we borrow from no experience, but which still lie in our representation a priori, are not mere phantasms of our brain.”

The pure intuitions of space and time play an important role in Kant’s philosophy of mathematics: they serve to justify the synthetic a priori status of geometry and arithmetic. When we judge that the sum of the interior angles of a triangle is 180º, for example, we do so not purely by examining the concepts triangle, sum, and angle. We also need to consult the pure intuition of space! And similarly, our affirmations of arithmetic truths rely upon the pure intuition of time for their validity.

Pure Concepts

Pure intuition is only one part of the story. We don’t just perceive the world, we also think about our perceptions. In Kant’s words, “Thoughts without content are empty; intuitions without concepts are blind. […] The understanding cannot intuit anything, and the senses cannot think anything. Only from their union can cognition arise.” As pure intuitions are to perceptions, pure concepts are to thought. Pure concepts are necessary for our empirical judgments, and without them we could not make sense of perception. It is this category in which causality falls.

Kant’s argument for this is that causality is a necessary condition for the judgment that events occur in a temporal order. He starts by observing that we don’t directly perceive time. For instance, we never have a perception of one event being before another, we just perceive one and, separately, the other. So to conclude that the first preceded the second requires something beyond perception, that is, a concept connecting them.

He argues that this connection must be necessary: “For this objective relation to be cognized as determinate, the relation between the two states must be thought as being such that it determines as necessary which of the states must be placed before and which after.” And as we’ve seen, the only way to get a necessary connection between perceptions is through a pure concept. The required pure concept is the relation of cause and effect: “the cause is what determines the effect in time, and determines it as the consequence.” So starting from the observation that we judge events to occur in a temporal order, Kant concludes that we must have a pure concept of cause and effect.

What about particular causal rules, like that striking a match produces a flame? Such rules are not derived solely from experience, but also from the pure concept of causality, on which their existence depends. It is the presence of the pure concept that allows the inference of these particular rules from experience, even though they postulate a necessary connection.

Now we can see how different Kant and Hume’s conceptions of causality are. While Hume thought that the traditional concept of causality as a necessary connection was unrescuable and extraneous to our perceptions, Kant sees it as a bedrock principle of experience that is necessary for us to be able to make sense of our perceptions at all. Kant rejects Hume’s definition of cause in terms of constant conjunction on the grounds that it “cannot be reconciled with the scientific a priori cognitions that we actually have.”

Despite this great gulf between the two philosophers’ conceptions of causality, there are some similarities. As we saw above, Kant agrees wholeheartedly with Hume that perception alone is insufficient for concluding that there is a necessary connection between events. He also agrees that a purely analytic approach is insufficient. Since Kant sees pure intuitions and pure concepts as features of the mind, not the external world, both philosophers deny that causation is an objective relationship between things in themselves (as opposed to perceptions of things). Of course, Kant would deny that this makes causality an illusion, just as he denied that space and time are made illusory by his philosophy.

Of course, it’s impossible to know to what extent the two philosophers would have actually agreed, had Hume been able to read Kant’s responses to his works. Would he have been convinced that synthetic a priori judgments really exist? If so, would he accept Kant’s pure intuitions and pure concepts? I suspect that at the crux of their disagreement would be Kant’s claim that math is synthetic a priori. While Hume never explicitly proclaims math’s analyticity (he didn’t have the term, after all), it seems more in line with his views on algebra and arithmetic as purely concerning the way that ideas relate to one another. It is also more in line with the axiomatic approach to mathematics familiar to Hume, in which one defines a set of axioms from which all truths about the mathematical concepts involved necessarily follow.

If Hume did maintain math’s analyticity, then Kant’s arguments about the importance of synthetic a priori knowledge would probably hold much less sway for him, and would largely amount to an appeal to the validity of metaphysical knowledge, which Hume already doubted. Hume also would likely want to resist Kant’s idealism; in Section XII of the Enquiry he mocks philosophers that doubt the connection between the objects of our senses and external objects, saying that if you “deprive matter of all its intelligible qualities, both primary and secondary, you in a manner annihilate it and leave only a certain unknown, inexplicable something as the cause of our perceptions – a notion so imperfect that no skeptic will think it worthwhile to contend against it.”

How to Learn From Data, Part 2: Evaluating Models

I ended the last post by saying that the solution to the problem of overfitting all relied on the concept of a model. So what is a model? Quite simply, a model is just a set of functions, each of which is a candidate for the true distribution that generates the data.

Why use a model? If we think about a model as just a set of functions, this seems kinda abstract and strange. But I think it can be made intuitive, and that in fact we reason in terms of models all the time. Think about how physicists formulate their theories. Laws of physics have two parts: First they specify the types of functional relationships there are in the world. And second, they specify the value of particular parameters in those functions. This is exactly how a model works!

Take Newton’s theory of gravity. The fundamental claim of this theory is that there is a certain functional relationship between the quantities a (the acceleration of an object), M (the mass of a nearby object), and r (the distance between the two objects): a ~ M/r2. To make this relationship precise, we need to include some constant parameter G that says exactly what this proportionality is: a = GM/r2.

So we start off with a huge set of probability distributions over observations of the accelerations of particles (one for each value of G), and then we gather data to see which value of G is most likely to be right. Einstein’s theory of gravity was another different model, specifying a different functional relationship between a, M, and r and involving its own parameters. So to compare Einstein’s theory of gravity to Newton’s is to compare different models of the phenomenon of gravitation.

When you start to think about it, the concept of a model is an enormous one. Models are ubiquitous, you can’t escape from them. Many fancy methods in machine learning are really just glorified model selection. For instance, a neural network is a model: the architecture of the network specifies a particular functional relationship between the input and the output, and the strengths of connections between the different neurons are the parameters of the model.

So. What are some methods for assessing predictive accuracy of models? The first one we’ll talk about is a super beautiful and clever technique called…

Cross Validation

The fundamental idea of cross validation is that you can approximate how well a model will do on the next data point by evaluating how the model would have done at predicting a subset of your data, if all it had access to was the rest of your data.

I’ll illustrate this with a simple example. Suppose you have a data set of four points, each of which is a tuple of two real numbers. Your two models are T1: that the relationship is linear, and T2: that the relationship is quadratic.

What we can do is go through each data point, selecting each in order, train the model on the non-selected data points, and then evaluate how well this trained model fits the selected data point. (By training the model, I mean something like “finding the curve within the model that best fits the non-selected data points.”) Here’s a sketch of what this looks like:

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There are a bunch of different ways of doing cross validation. We just left out one data point at a time, but we could have left out two points at a time, or three, or any k less than the total number of points N. This is called leave-k-out cross validation. If we partition our data by choosing a fraction of it for testing, then we get what’s called n-fold cross validation (where 1/n is the fraction of the data that is isolated for testing).

We also have some other choices besides how to partition the data. For instance, we can choose to train our model via a Likelihoodist procedure or a Bayesian procedure. And we can also choose to test our model in various ways, by using different metrics for evaluating the distance between the testing set and the trained model. In leave-one-out cross validation (LOOCV), a pretty popular method, both the training and testing procedures are Likelihoodist.

Now, there’s a practical problem with cross-validation, which is that it can take a long time to compute. If we do LOOCV with N data points and look at all ways of leaving out one data point, then we end up doing N optimization procedures (one for each time you train your model), each of which can be pretty slow. But putting aside practical issues, for thinking about theoretical rationality, this is a super beautiful technique.

Next we’ll move on to…

Bayesian Model Selection!

We had Bayesian procedures for evaluating individual functions. Now we can go full Bayes and apply it to models! For each model, we assess the posterior probability of the model given the data using Bayes’ rule as usual:

Pr(M | D) = \frac{Pr(D | M)}{Pr(D)} Pr(M)

Now… what is the prior Pr(M)? Again, it’s unspecified. Maybe there’s a good prior that solves overfitting, but it’s not immediately obvious what exactly it would be.

But there’s something really cool here. In this equation, there’s a solution to overfitting that pops up not out of the prior, but out of the LIKELIHOOD! I explain this here. The general idea is that models that are prone to overfitting get that way because their space of functions is very large. But if the space of functions is large, then the average prior on each function in the model must be small. So larger models have, on average, smaller values of the term Pr(f | M), and correspondingly (via Pr(D | M) = \sum_{f \in M} Pr(D | f, M) Pr(f | M) ) get a weaker update from evidence.

So even without saying anything about the prior, we can see that Bayesian model selection provides a potential antidote to overfitting. But just like before, we have the practical problem that computing Pr(M | D) is in general very hard. Usually evaluating Pr(D | M) involves calculating a really complicated many-dimensional integral, and calculating many-dimensional integrals can be very computationally expensive.

Might we be able to find some simple unbiased estimator of the posterior Pr(M | D)? It turns out that yes, we can. Take another look at the equation above.

Pr(M | D) = \frac{Pr(D | M)}{Pr(D)} Pr(M)

Since Pr(D) is a constant across models, we can ignore it when comparing models. So for our purposes, we can attempt to maximize the equation:

Pr(D | M) Pr(M) = \sum_{f \in M} Pr(D | f, M) Pr(f | M) Pr(M)

If we assume that P(D | f) is twice differentiable in the model parameters and that Pr(f) behaves roughly linearly around the maximum likelihood function, we can approximate this as:

Pr(D | M) Pr(M) \approx \frac {Pr(D | f^*(D))} {\sqrt{N}}^k Pr(M)

f* is the maximum likelihood function within the model, and k is the the number of parameters of the model (e.g. k = 2 for a linear model and k = 3 for a quadratic model).

\We can make this a little neater by taking a logarithm and defining a new quantity: the Bayesian Information Criterion.

argmax_M [ Pr(M | D) ] \\~\\  = argmax_M [ Pr(D | M) Pr(M) ] \\~\\  \approx argmax_M [ \frac {Pr(D | f^*(D))} {\sqrt{N}}^k Pr(M)] \\~\\  = argmax_M [ \log(Pr(M)) + \log (Pr(D | f^*(D)) - \frac{k}{2} \log(N) ] \\~\\  = argmin_M [ BIC - \log Pr(M) ]

BIC = \frac{k}{2} log(N) - \log \left( Pr(D | f^*(D)) \right)

Thus, to maximize the posterior probability of a model M is roughly the same as to minimize the quantity BIC – log Pr(M).

If we assume that our prior over models is constant (that is, that all models are equally probable at the outset), then we just minimize BIC.

Bayesian Information Criterion

Notice how incredibly simple this expression is to evaluate! If all we’re doing is minimizing BIC, then we only need to find the maximum likelihood function f*(D), assess the value of Pr(D | f*), and then penalize this quantity with the factor k/2 log(N)! This penalty scales in proportion to the model complexity, and thus helps us avoid overfitting.

We can think about this as a way to make precise the claims above about Bayesian Model Selection penalizing overfitting in the likelihood. Remember that minimizing BIC made sense when we assumed a uniform prior over models (and therefore when our prior doesn’t penalize overfitting). So even when we don’t penalize overfitting in the prior, we still end up getting a penalty! This penalty must come from the likelihood.

Some more interesting facts about BIC:

  • It is approximately equal to the minimum description length criterion (which I haven’t discussed here)
  • It is only valid when N >> k. So it’s not good for small data sets or large models.
  • It the truth is contained within your set of models, then BIC will select the truth with probability 1 as N goes to infinity.

Okay, stepping back. We sort of have two distinct clusters of approaches to model selection. There’s Bayesian Model Selection and the Bayesian Information Criterion, and then there’s Cross Validation. Both sound really nice. But how do they compare? It turns out that in general they give qualitatively different answers. And in general, the answers you get using cross validation tend to be more predictively accurate that those that you get from BIC/BMS.

A natural question to ask at this point is: BIC was a nice simple approximation to BMS. Is there a corresponding nice simple approximation to cross validation? Well first we must ask: which cross validation? Remember that there was a plethora of different forms of cross validation, each corresponding to slightly different criterion for evaluating fits. We can’t assume that all these methods give the same answer.

Let’s choose leave-one-out cross validation. It turns out that yes, there is a nice approximation that is asymptotically equivalent to LOOCV! This is called the Akaike information criterion.

Akaike Information Criterion

First, let’s define AIC:

AIC = k - \log \left( Pr(D | f^*(D)) \right)

Like always, k is the number of parameters in M and f* is chosen by the Likelihoodist approach described in the last post.

What you get by minimizing this quantity is asymptotically equivalent to what you get by doing leave-one-out cross validation. Compare this to BIC:

BIC = \frac{k}{2} \log (N) - \log \left( Pr(D | f^*(D) ) \right)

There’s a qualitative difference in how the parameter penalty is weighted! BIC is going to have a WAY higher complexity penalty than AIC. This means that AIC should in general choose less simple models than BIC.

Now, we’ve already seen one reason why AIC is good: it’s a super simple approximation of LOOCV and LOOCV is good. But wait, there’s more! AIC can be derived as an unbiased estimator of the Kullback-Leibler divergence DKL.

What is DKL? It’s a measure of the information theoretic distance of a model from truth. For instance, if the true generating distribution for a process is f, and our model of this process is g, then DKL tells us how much information we lose by using g to represent f. Formally, DKL is:

D_{KL}(f, g) = \int { f(x) \log \left(\frac{f(x)} {g(x)} \right) dx }

Notice that to actually calculate this quantity, you have to already know what the true distribution f is. So that’s unfeasible. BUT! AIC gives you an unbiased estimate of its value! (The proof of this is complicated, and makes the assumptions that N >> k, and that the model is not far from the truth.)

Thus, AIC gives you an unbiased estimate of which model is closest to the truth. Even if the truth is not actually contained within your set of models, what you’ll end up with is the closest model to it. And correspondingly, you’ll end up getting a theory of the process that is very similar to how the process actually works.

There’s another version of AIC that works well for smaller sample sizes and larger models called AICc.

AICc = AIC + \frac {k (k + 1)} {N - (k + 1)}

And interestingly, AIC can be derived as an approximation to Bayesian Model Selection with a particular prior over models. (Remember earlier when we showed that argmax_M \left( Pr(M | D) \right) \approx argmin_M \left( BIC - \log(Pr(M)) \right) ? Well, just set \log Pr(M) = BIC - AIC = k \left( \frac{1}{2} \log N - 1 \right) , and you get the desired result.) Interestingly, this prior ends up rewarding theories for having lots of parameters! This looks pretty bad… it seems like AIC is what you get when you take Bayesian Model Selection and then try to choose a prior that favors overfitting theories. But the practical use and theoretical virtues of AIC warrant, in my opinion, taking a closer look at what’s going on here. Perhaps what’s going on is that the likelihood term Pr(D | M) is actually doing too much to avoid overfitting, so in the end what we need in our prior is one that avoids underfitting! Regardless, we can think of AIC as specifying the unique good prior on models that optimizes for predictive accuracy.

There’s a lot more to be said from here. This is only a short wade into the waters of statistical inference and model selection. But I think this is a good place to stop, and I hope that I’ve given you a sense of the philosophical richness of the various different frameworks for inference I’ve presented here.

How to Learn From Data, Part I: Evaluating Simple Hypotheses

Here’s a general question of great importance: How should we adjust our model of the world in response to data? This post is a tour of a few of the big ideas that humans have come up with to address this question. I find learning about this topic immensely rewarding, and so I shall attempt to share it! What I love most of all is thinking about how all of this applies to real world examples of inference from data, so I encourage you to constantly apply what I say to situations that you find personally interesting.

First of all, we want a formal description of what we mean by data. Let’s describe our data quite simply as a set: D = \{(x_1, y_1), (x_2, y_2),..., (x_N, y_N)\}, where each x_i is a value that you choose (your independent variable) and each y_i is the resulting value that you measure (the dependent variable). Each of these variables can be pretty much anything – real numbers, n-tuples of real numbers, integers, colors, whatever you want. The goal here is to embrace generality, so as to have a framework that applies to many kinds of inference.

Now, suppose that you have a certain theory about the underlying relationship between the x variables and the y variables. This theory might take the form of a simple function: T: y = f(x) We interpret T as making a prediction of a particular value of the dependent variable for each value of the independent variable. Maybe the data is the temperatures of regions at various altitudes, and our theory T says that one over the temperature (1/T) is some particular linear function of the altitude.

What we want is a notion of how good of a theory T is, given our data. Intuitively, we might think about doing this by simply assessing the distance between each data point y_n and the predicted value of y at that point: f(x_n), using some metric, then adding them all up. But there are a whole bunch of distance metrics available to us. Which one should we use? Perhaps the taxicab measure comes to mind (\sum_{n=1}^N {|y_n - f(x_n)|}), or the sum of the squares of the differences (SOS = \sum_{n=1}^N {(y_n - f(x_n))^2}). We want a good theoretical justification for why any one of these metrics should be preferred over any other, since in general they lead to different conclusions. We’ll see just such justifications shortly. Keep the equation for SOS in mind, as it will turn up repeatedly ahead.

Now here’s a problem: If we want a probabilistic evaluation of T, then we have to face the fact that it makes a deterministic prediction. Our theory seems to predict with 100% probability that at x_n the observed y_n will be precisely f(x_n). If it’s even slightly off of this, then our theory will be probabilistically disconfirmed to zero.

We can solve this problem by modifying our theory T to not just be a theory of the data, but also a theory of error. In other words, we expect that the value we get will not be exactly the predicted value, and give some account of how on average observation should differ from theory.

T: y = f(x) + \epsilon, where \epsilon is some random variable drawn from a probability distribution P_E.

This error distribution can be whatever we please – Gaussian, exponential, Poisson, whatever. For simplicity let’s say that we know the error is normal (drawn from a Gaussian distribution) with a known standard deviation σ.

T: y = f(x) + \epsilon, \epsilon \sim N(0, \sigma)

A note on notation here: \epsilon \sim N(\mu, \sigma) denotes a random variable drawn from a Gaussian distribution centered around \mu with a standard deviation of \sigma.

This gives us a sensible notion of the probability of obtaining some value of y from a chosen x given the theory T.

Pr(y | x, T) \sim N(f(x), \sigma) \\~\\  Pr(y | x, T) = \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{1}{2 \sigma^2} (y - f(x))^2} 

Nice! This is a crucial step towards figuring out how to evaluate theories: we’ve developed a formalism for describing precisely how likely a data point is, given a particular theory (which, remember, so far is just a function from values of independent variables to values of dependent variables, coupled with a theory of how the error in your observations works).

Let’s try to extend this to our entire data set D. We want to assess the probability of the particular values of the dependent variables, given the chosen values of the dependent variables and the function. We’ll call this the probability of D given f.

Pr(D | f) = Pr(x_1, y_1, x_2, y_2,..., x_N, y_N | T) \\~\\  Pr(D | f) = Pr(y_1, y_2,...,y_N | x_1,  x_2,..., x_N, T)

(It’s okay to move the values of the independent variable x across the conditional bar because of our assumption that we know beforehand what values x will take on.)

But now we run into a problem: we can’t really do anything with this expression without knowing how the data points depend upon each other. We’d like to break it into individual terms, each of which can be evaluated by the above expression for Pr(y | x, T). But that would require an assumption that our data points are independent. In general, we cannot grant this assumption. But what we can do is expand our theory once more to include a theory of the dependencies in our data points. For simplicity of explication, let’s proceed with the assumption that each of our observations is independent of the others. Said another way, we assume that given T, x_n screens off all other variables from y_n). Combined with our assumption of normal error, this gives us a nice simple reduction.

Pr(D | f) = Pr(y_1, y_2,...,y_N | x_1,  x_2,..., x_N, T) \\~\\  Pr(D | f) = Pr(y_1 | x_1, T) Pr(y_2 | x_2, T) ... Pr(y_N | x_N, T) \\~\\  Pr(D | f) = \prod_{n=1}^N { \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{1}{2 \sigma^2} (y_n - f(x_n))^2} } \\~\\  Pr(D | f) = (2 \pi \sigma^2)^{-N/2} e^{-\frac{1}{2 \sigma^2} \sum_{n=1}^N {(y_n - f(x_n))^2}} \\~\\  Pr(D | f) = (2 \pi \sigma^2)^{-N/2} e^{-\frac{SOS(f, D)}{2 \sigma^2} }

We see an interesting connection arise here between Pr(D | f) and the sum of squares evaluation of fit. More will be said of this in just a moment.

But first, let’s take a step back. Our ultimate goal here is to find a criterion for theory evaluation. And here we’ve arrived at an expression that looks like it might be right for that role! Perhaps we want to say that our criterion for evaluating theories is just maximization of Pr(D | f). This makes some intuitive sense… a good theory is one which predicts the observed data. If general relativity or quantum mechanics predicted that the sun should orbit the earth, or that atoms should be unstable, then we wouldn’t be very in favor of it.

And so we get the Likelihoodism Thesis!

Likelihoodism: The best theory f* given some data D is that which maximizes Pr(D | f). Formally: f^*(D) = argmax_f [ Pr(D | f) ]

Now, since logarithms are monotonic, we can express this in a more familiar form:

argmax_f [Pr(D | F)] = argmax_f \left[ e^{-\frac{SOS(f, D)}{2 \sigma^2} } \right] = argmax_f \left[-\frac{SOS(f, D)}{2 \sigma^2} \right] = argmin_f [ SOS(f, D) ]

Thus the best theory according to Likelihoodism is that which minimizes the sum of squares! People minimize sums of squares all the time, and most of the time they don’t realize that it can be derived from the Likelihoodism Thesis and the assumptions of Gaussian error and independent data. If our assumptions had been different, then we would have found a different expression, and SOS might no longer be appropriate! For instance, the taxicab metric arises as the correct metric if we assume exponential error rather than normal error. I encourage you to see why this is for yourself. It’s a fun exercise to see how different assumptions about the data give rise to different metrics for evaluating the distance between theory and observation. In general, you should now be able to take any theory of error and assess what the proper metric for “degree of fit of theory to data” is, assuming that degree of fit is evaluated by maximizing Pr(D | f).

Now, there’s a well-known problem with the Likelihoodism thesis. This is that the function f that minimizes SOS(f, D) for some data D is typically going to be a ridiculously overcomplicated function that perfectly fits each data points, but does terribly on future data. The function will miss the underlying trend in the data for the noise in the observations, and as a result fail to get predictive accuracy.

This is the problem of overfitting. Likelihoodism will always prefer theories that overfit to those that don’t, and as a result will fail to identify underlying patterns in data and do a terrible job at predicting future data.

How do we solve this? We need a replacement for the Likelihoodism thesis. Here’s a suggestion: we might say that the problem stems from the fact that the Likelihoodist procedure recommends us to find a function that makes the data most probable, rather than finding a function that is made most probable by the data. From this suggestion we get the Bayesian thesis:

Bayesianism: The best theory f given some data is that which maximizes Pr(f | D). f^*(D) = argmax_f [ Pr(f | D) ]

Now, what is this Pr(f | D) term? It’s an expression that we haven’t seen so far. How do we evaluate it? We simply use the theorem that the thesis is named for: Bayes’ rule!

Pr(f | D) = \frac{Pr(D | f)}{Pr(D)} Pr(f)

This famous theorem is a simple deductive consequence of the probability axioms and the definition of conditional probability. And it happens to be exactly what we need here. Notice that the right-hand side consists of the term Pr(D | f), which we already know how to calculate. And since we’re ultimately only interested in varying f to find the function that maximizes this expression, we can ignore the constant term in the denominator.

argmax_f [ Pr(f | D) ] = argmax_f [ Pr(D | f) Pr(f) ] \\~\\  argmax_f [ Pr(f | D) ] = argmax_f [ \log Pr(D | f) + \log Pr(f) ] \\~\\  argmax_f [ Pr(f | D) ] = argmax_f [ -\frac{SOS(f, D)}{2 \sigma^2} + \log Pr(f) ] \\~\\  argmax_f [ Pr(f | D) ] = argmin_f [ SOS(f, D) - 2 \sigma^2 \log Pr(f) ]

The last steps we get just by substituting in what we calculated before. The 2σ² in the first term comes from the fact that the exponent of our Gaussian is (y – f(x))² / 2σ². We ignored it before because it was just a constant, but since we now have another term in the expression being maximized, we have to keep track of it again.

Notice that what we get is just what we had initially (a sum of squares) plus an additional term involving a mysterious Pr(f). What is this Pr(f)? It’s our prior distribution over theories. Because of the negative sign in front of the second term, a larger value of Pr(f) gives a smaller value for the expression that we are minimizing. Similarly, the more closely our function follows the data, the smaller the SOS term becomes. So what we get is a balance between fitting the data and having a high prior. A theory that fits the data perfectly can still end up with a bad evaluation, as long as it has a low enough prior. And a theory that fits the data poorly can end up with a great evaluation if it has a high enough prior.

(Those that are familiar with machine learning might notice that this feels similar to regularization. They’re right! It turns out that different regularization techniques just end up corresponding to different Bayesian priors! L2 regularization corresponds to a Gaussian prior over parameters, and L1 regularization corresponds to an exponential prior over parameters. And so on. Different prior assumptions about the process generating the data lead naturally to different regularization techniques.)

What this means is that if we are trying to solve overfitting, then the Bayesianism thesis provides us a hopeful ray of light. Maybe we can find some perfect prior Pr(f) that penalizes complex overfitting theories just enough to give us the best prediction. But unsurprisingly, finding such a prior is no easy matter. And importantly, the Bayesian thesis as I’ve described it gives us no explicit prescription for what this prior should be, making it insufficient for a full account of inference. What Bayesianism does is open up a space of possible methods of theory evaluation, inside which (we hope) there might be a solution to our problem.

Okay, let’s take another big step back. How do we progress from here? Well, let’s think for a moment about what our ultimate goal is. We want to say that overfitting is bad, and that therefore some priors are better than others insofar as they prevent overfitting. But what is the standard we’re using to determine that overfitting is bad? What’s wrong with an overfitting theory?

Here’s a plausible answer: The problem with overfitting is that while it maximizes descriptive accuracy, it leads to poor predictive accuracy! I.e., theories that overfit do a great job at describing past data, but tend to do a poor job of matching future data.

Let’s take this idea and run with it. Perhaps all that we truly care about in theory selection is predictive accuracy. I’ll give a name to this thesis:

Predictivism: The ultimate standard for theory evaluation is predictive accuracy. I.e., the best theory is that which does the best at predicting future data.

Predictivism is a different type of thesis than Bayesianism and Likelihoodism. While each of those gave precise prescriptions for how to calculate the best theory, Predictivism does not. If we want an explicit algorithm for computing the best theory, then we need to say what exactly “predictive accuracy” means. This means that as far as we know so far, Predictivism could be equivalent to Likelihoodism or to Bayesianism. It all depends on what method we end up using to determine what theory has the greatest predictive accuracy. Of course, we have good reasons to suspect that Likelihoodism is not identical to Predictivism (Likelihoodism overfits, and we suspect that Predictivism will not) or to Bayesianism (Bayesianism does not prescribe a particular prior, so it gives no unique prescription for the best theory). But what exactly Predictivism does say depends greatly on how exactly we formalize the notion of predictive accuracy.

Another difference between Predictivism and Likelihoodism/Bayesianism is that predictive accuracy is about future data. While it’s in principle possible to find an analytic solution to P(D | f) or P(f | D), Predictivism seems to require us to compute terms involving future data! But these are impossible to specify, because we don’t know what the future data will be!

Can we somehow approximate what the best theory is according to Predictivism, by taking our best guess at what the future data will be? Maybe we can try to take the expected value of something like Pr(y_{N+1} | x_{N+1}, T, D) , where (x_{N+1}, y_{N+1}) is a future data point. But there are two big problems with this.

First, in taking an expected value, you must use the distribution of outcomes given by your own theory. But then your evaluation picks up an inevitable bias! You can’t rely on your own theories to assess how good your theories are.

And second, by the assumptions we’ve included in our theories so far, the future data will be independent of the past data! This is a really big problem. We want to use our past data to give some sense of how well we’ll do on future data. But if our data is independent, then our assessment of how well T will do on the next data point will have to be independent of the past data as well! After all, no matter what value y_{N+1} has, Pr(y_{N+1} | x_{N+1}, T, D) = Pr(y_{N+1} | x_{N+1}, T).

So is there any way out of this? Surprisingly, yes! If we get creative, we can find ways to approximate this probability with some minimal assumptions about the data. These methods all end up relying on a new concept that we haven’t yet discussed: the concept of a model. The next post will go into more detail on how this works. Stay tuned!

Gödel’s Second Incompleteness Theorem: Explained in Words of Only One Syllable

Somebody recently referred me to a 1994 paper by George Boolos in which he writes out a description of Gödel’s Second Incompleteness Theorem, using only words of one syllable. I love it so much that I’m going to copy the whole thing here in this post. Enjoy!

First of all, when I say “proved”, what I will mean is “proved with the aid of the whole of math”. Now then: two plus two is four, as you well know. And, of course, it can be proved that two plus two is four (proved, that is, with the aid of the whole of math, as I said, though in the case of two plus two, of course we do not need the whole of math to prove that it is four). And, as may not be quite so clear, it can be proved that it can be proved that two plus two is four, as well. And it can be proved that it can be proved that it can be proved that two plus two is four. And so on. In fact, if a claim can be proved, then it can be proved that the claim can be proved. And that too can be proved.

Now, two plus two is not five. And it can be proved that two plus two is not five. And it can be proved that it can be proved that two plus two is not five, and so on.

Thus: it can be proved that two plus two is not five. Can it be proved as well that two plus two is five? It would be a real blow to math, to say the least, if it could. If it could be proved that two plus two is five, then it could be proved that five is not five, and then there would be no claim that could not be proved, and math would be a lot of bunk.

So, we now want to ask, can it be proved that it can’t be proved that two plus two is five? Here’s the shock: no, it can’t. Or, to hedge a bit: if it can be proved that it can’t be proved that two plus two is five, then it can be proved as well that two plus two is five, and math is a lot of bunk. In fact, if math is not a lot of bunk, then no claim of the form “claim X can’t be proved” can be proved.

So, if math is not a lot of bunk, then, though it can’t be proved that two plus two is five, it can’t be proved that it can’t be proved that two plus two is five.

By the way, in case you’d like to know: yes, it can be proved that if it can be proved that it can’t be proved that two plus two is five, then it can be proved that two plus two is five.

George Boolos, Mind, Vol. 103, January 1994, pp. 1 – 3

Anti-inductive priors

I used to think of Bayesianism as composed of two distinct parts: (1) setting priors and (2) updating by conditionalizing. In my mind, this second part was the crown jewel of Bayesian epistemology, while the first part was a little more philosophically problematic. Conditionalization tells you that for any prior distribution you might have, there is a unique rational set of new credences that you should adopt upon receiving evidence, and tells you how to get it. As to what the right priors are, well, that’s a different story. But we can at least set aside worries about priors with assurances about how even a bad prior will eventually be made up for in the long run after receiving enough evidence.

But now I’m realizing that this framing is pretty far off. It turns out that there aren’t really two independent processes going on, just one (and the philosophically problematic one at that): prior-setting. Your prior fully determines what happens when you update by conditionalization on any future evidence you receive. And the set of priors consistent with the probability axioms is large enough that it allows for this updating process to be extremely irrational.

I’ll illustrate what I’m talking about with an example.

Let’s imagine a really simple universe of discourse, consisting of just two objects and one predicate. We’ll make our predicate “is green” and denote objects a_1 and a_2 . Now, if we are being good Bayesians, then we should treat our credences as a probability distribution over the set of all state descriptions of the universe. These probabilities should all be derivable from some hypothetical prior probability distribution over the state descriptions, such that our credences at any later time are just the result of conditioning that prior on the total evidence we have by that time.

Let’s imagine that we start out knowing nothing (i.e. our starting credences are identical to the hypothetical prior) and then learn that one of the objects (a_1 ) is green. In the absence of any other information, then by induction, we should become more confident that the other object is green as well. Is this guaranteed by just updating?

No! Some priors will allow induction to happen, but others will make you unresponsive to evidence. Still others will make you anti-inductive, becoming more and more confident that the next object is not green the more green things you observe. And all of this is perfectly consistent with the laws of probability theory!

Take a look at the following three possible prior distributions over our simple language:

Screen Shot 2018-10-21 at 1.58.45 PM.png

According to P_1 , your new credence in Ga_2 after observing Ga_1 is P_1(Ga_2 | Ga_1) = 0.80 , while your prior credence in Ga_2 was 0.50. Thus P_1 is an inductive prior; you get more confident in future objects being green when you observe past objects being green.

For P_2 , we have that P_2(Ga_2 | Ga_1) = 0.50 , and P_2(Ga_2) = 0.50 as well. Thus P_2 is a non-inductive prior: observing instances of green things doesn’t make future instances of green things more likely.

And finally, P_3(Ga_2 | Ga_1) = 0.20 , while P_3(Ga_2) = 0.5 . Thus P_3 is an anti-inductive prior. Observing that one object is green makes you more than two times less confident confident that the next object will be green.

The anti-inductive prior can be made even more stark by just increasing the gap between the prior probability of Ga_1 \wedge Ga_2 and Ga_1 \wedge -Ga_2 . It is perfectly consistent with the axioms of probability theory for observing a green object to make you almost entirely certain that the next object you observe will not be green.

Our universe of discourse here was very simple (one predicate and two objects). But the point generalizes. Regardless of how many objects and predicates there are in your language, you can have non-inductive or anti-inductive priors. And it isn’t even the case that there are fewer anti-inductive priors than inductive priors!

The deeper point here is that the prior is doing all the epistemic work. Your prior isn’t just an initial credence distribution over possible hypotheses, it also dictates how you will respond to any possible evidence you might receive. That’s why it’s a mistake to think of prior-setting and updating-by-conditionalization as two distinct processes. The results of updating by conditionalization are determined entirely by the form of your prior!

This really emphasizes the importance of having good criterion for setting priors. If we’re trying to formalize scientific inquiry, it’s really important to make sure our formalism rules out the possibility of anti-induction. But this just amounts to requiring rational agents to have constraints on their priors that go above and beyond the probability axioms!

What are these constraints? Do they select one unique best prior? The challenge is that actually finding a uniquely rationally justifiable prior is really hard. Carnap tried a bunch of different techniques for generating such a prior and was unsatisfied with all of them, and there isn’t any real consensus on what exactly this unique prior would be. Even worse, all such suggestions seem to end up being hostage to problems of language dependence – that is, that the “uniquely best prior” changes when you make an arbitrary translation from your language into a different language.

It looks to me like our best option is to abandon the idea of a single best prior (and with it, the notion that rational agents with the same total evidence can’t disagree). This doesn’t have to lead to total epistemic anarchy, where all beliefs are just as rational as all others. Instead, we can place constraints on the set of rationally permissible priors that prohibit things like anti-induction. While identifying a set of constraints seems like a tough task, it seems much more feasible than the task of justifying objective Bayesianism.

Making sense of improbability

Imagine that you take a coin that you believe to be fair and flip it 20 times. Each time it lands heads. You say to your friend: “Wow, what a crazy coincidence! There was a 1 in 220 chance of this outcome. That’s less than one in a million! Super surprising.”

Your friend replies: “I don’t understand. What’s so crazy about the result you got? Any other possible outcome (say, HHTHTTTHTHHHTHTTHHHH) had an equal probability as getting all heads. So what’s so surprising?”

Responding to this is a little tricky. After all, it is the case that for a fair coin, the probability of 20 heads = the probability of HHTHTTTHTHHHTHTTHHHH = roughly one in a million.

Simpler Example_ Five Tosses.png

So in some sense your friend is right that there’s something unusual about saying that one of these outcomes is more surprising than another.

You might answer by saying “Well, let’s parse up the possible outcomes by the number of heads and tails. The outcome I got had 20 heads and 0 tails. Your example outcome had 12 heads and 8 tails. There are many many ways of getting 12 heads and 8 tails than of getting 20 heads and 0 tails, right? And there’s only one way of getting all 20 heads. So that’s why it’s so surprising.”

Probability vs. Number of heads (1).png

Your friend replies: “But hold on, now you’re just throwing out information. Sure my example outcome had 12 heads and 8 tails. But while there’s many ways of getting that number of heads and tails, there’s only exactly one way of getting the result I named! You’re only saying that your outcome is less likely because you’ve glossed over the details of my outcome that make it equally unlikely: the order of heads and tails!”

I think this is a pretty powerful response. What we want is a way to say that HHHHHHHHHHHHHHHHHHHH is surprising while HHTHTTTHTHHHTHTTHHHH is not, not that 20 heads is surprising while 12 heads and 8 tails is unsurprising. But it’s not immediately clear how we can say this.

Consider the information theoretic formalization of surprise, in which the surprisingness of an event E is proportional to the negative log of the probability of that event: Sur(E) = -log(P(E)). There are some nice reasons for this being a good definition of surprise, and it tells us that two equiprobable events should be equally surprising. If E is the event of observing all heads and E’ is the event of observing the sequence HHTHTTTHTHHHTHTTHHHH, then P(E) = P(E’) = 1/220. Correspondingly, Sur(E) = Sur(E’). So according to one reasonable formalization of what we mean by surprisingness, the two sequences of coin tosses are equally surprising. And yet, we want to say that there is something more epistemically significant about the first than the second.

(By the way, observing 20 heads is roughly 6.7 times more surprising than observing 12 heads and 8 tails, according to the above definition. We can plot the surprise curve to see how maximum surprise occurs at the two ends of the distribution, at which point it is 20 bits.)

Surprise vs. number of heads (1).png

So there is our puzzle: in what sense does it make sense to say that observing 20 heads in a row is more surprising than observing the sequence HHTHTTTHTHHHTHTTHHHH? We certainly have strong intuitions that this is true, but do these intuitions make sense? How can we ground the intuitive implausibility of getting 20 heads? In this post I’ll try to point towards a solution to this puzzle.

Okay, so I want to start out by categorizing three different perspectives on the observed sequence of coin tosses. These correspond to (1) looking at just the outcome, (2) looking at the way in which the observation affects the rest of your beliefs, and (3) looking at how the observation affects your expectation of future observations. In probability terms, these correspond to the P(E), P(T| T) and P(E’ | E).

Looking at things through the first perspective, all outcomes are equiprobable, so there is nothing more epistemically significant about one than the other.

But considering the second way of thinking about things, there can be big differences in the significance of two equally probable observations. For instance, suppose that our set of theories under consideration are just the set of all possible biases of the coin, and our credences are initially peaked at .5 (an unbiased coin). Observing HHTHTTTHTHHHTHTTHHHH does little to change our prior. It shifts a little bit in the direction of a bias towards heads, but not significantly. On the other hand, observing all heads should have a massive effect on your beliefs, skewing them exponentially in the direction of extreme heads biases.

Importantly, since we’re looking at beliefs about coin bias, our distributions are now insensitive to any details about the coin flip beyond the number of heads and tails! As far as our beliefs about the coin bias go, finding only the first 8 to be tails looks identical to finding the last 8 to be tails. We’re not throwing out the information about the particular pattern of heads and tails, it’s just become irrelevant for the purposes of consideration of the possible biases of the coin.

Visualizing change in beliefs about coin bias.png

If we want to give a single value to quantify the difference in epistemic states resulting from the two observations, we can try looking at features of these distributions. For instance, we could look at the change in entropy of our distribution if we see E and compare it to the change in entropy upon seeing E’. This gives us a measure of how different observations might affect our uncertainty levels. (In our example, observing HHTHTTTHTHHHTHTTHHHH decreases uncertainty by about 0.8 bits, while observing all heads decreases uncertainty by 1.4 bits.) We could also compare the means of the posterior distributions after each observation, and see which is shifted most from the mean of the prior distribution. (In this case, our two means are 0.57 and 0.91).

Now, this was all looking at things through what I called perspective #2 above: how observations affect beliefs. Sometimes a more concrete way to understand the effect of intuitively implausible events is to look at how they affect specific predictions about future events. This is the approach of perspective #3. Sticking with our coin, we ask not about the bias of the coin, but about how we expect it to land on the next flip. To assess this, we look at the posterior predictive distributions for each posterior:

Posterior Predictive Distributions.png

It shouldn’t be too surprising that observing all heads makes you more confident that the next coin will land heads than observing HHTHTTTHTHHHTHTTHHHH. But looking at this graph gives a precise answer to how much more confident you should be. And it’s somewhat easier to think about than the entire distribution over coin biases.

I’ll leave you with an example puzzle that relates to anthropic reasoning.

Say that one day you win the lottery. Yay! Super surprising! What an improbable event! But now compare this to the event that some stranger Bob Smith wins the lottery. This doesn’t seem so surprising. But supposing that Bob Smith buys lottery tickets at the same rate as you, the probability that you win is identical to the probability that Bob Smith wins. So… why is it any more surprising when you win?

This seems like a weird question. Then again, so did the coin-flipping question we started with. We want to respond with something like “I’m not saying that it’s improbable that some random person wins the lottery. I’m interested in the probability of me winning the lottery. And if we parse up the outcomes as that either I win the lottery or that somebody else wins the lottery, then clearly it’s much more improbable that I win than that somebody else wins.”

But this is exactly parallel to the earlier “I’m not interested in the precise sequence of coin flips, I’m just interested in the number of heads versus tails.” And the response to it is identical in form: If Bob Smith, a particular individual whose existence you are aware of, wins the lottery and you know it, then it’s cheating to throw away those details and just say “Somebody other than me won the lottery.” When you update your beliefs, you should take into account all of your evidence.

Does the framework I presented here help at all with this case?