Technical Report IDSIA0306
On the Foundations of
Universal Sequence Prediction
Abstract
Solomonoff completed the Bayesian framework by providing a rigorous, unique, formal, and universal choice for the model class and the prior. We discuss in breadth how and in which sense universal (noni.i.d.) sequence prediction solves various (philosophical) problems of traditional Bayesian sequence prediction. We show that Solomonoff’s model possesses many desirable properties: Fast convergence and strong bounds, and in contrast to most classical continuous prior densities has no zero p(oste)rior problem, i.e. can confirm universal hypotheses, is reparametrization and regrouping invariant, and avoids the oldevidence and updating problem. It even performs well (actually better) in noncomputable environments.
Contents
Keywords
Sequence prediction, Bayes, Solomonoff prior, Kolmogorov complexity, Occam’s razor, prediction bounds, model classes, philosophical issues, symmetry principle, confirmation theory, reparametrization invariance, oldevidence/updating problem, (non)computable environments.
1 Introduction
Examples and goal. Given the weather in the past, what is the probability of rain tomorrow? What is the correct answer in an IQ test asking to continue the sequence 1,4,9,16,? Given historic stockcharts, can one predict the quotes of tomorrow? Assuming the sun rose 5000 years every day, how likely is doomsday (that the sun does not rise) tomorrow? These are instances of the important problem of inductive inference or timeseries forecasting or sequence prediction. Finding prediction rules for every particular (new) problem is possible but cumbersome and prone to disagreement or contradiction. What we are interested in is a formal general theory for prediction.
Bayesian sequence prediction. The Bayesian framework is the most consistent and successful framework developed thus far [Ear93]. A Bayesian considers a set of environments=hypotheses=models which includes the true data generating probability distribution . From one’s prior belief in environment and the observed data sequence , Bayes’ rule yields one’s posterior confidence in . In a predictive setting, one directly determines the predictive probability of the next symbol without the intermediate step of identifying a (true or good or causal or useful) model. Note that classification and regression can be regarded as special sequence prediction problems, where the sequence of pairs is given and the class label or function value shall be predicted.
Universal sequence prediction. The Bayesian framework leaves open how to choose the model class and prior . General guidelines are that should be small but large enough to contain the true environment , and should reflect one’s prior (subjective) belief in or should be noninformative or neutral or objective if no prior knowledge is available. But these are informal and ambiguous considerations outside the formal Bayesian framework. Solomonoff’s [Sol64] rigorous, essentially unique, formal, and universal solution to this problem is to consider a single large universal class suitable for all induction problems. The corresponding universal prior is biased towards simple environments in such a way that it dominates=superior to all other priors. This leads to an a priori probability which is equivalent to the probability that a universal Turing machine with random input tape outputs .
History and motivation. Many interesting, important, and deep results have been proven for Solomonoff’s universal distribution [ZL70, Sol78, LV97, Hut04]. The motivation and goal of this paper is to provide a broad discussion of how and in which sense universal sequence prediction solves all kinds of (philosophical) problems of Bayesian sequence prediction, and to present some recent results. Many arguments and ideas could be further developed. I hope that the exposition stimulates such a future, more detailed, investigation.
Contents. In Section 2 we review the excellent predictive performance of Bayesian sequence prediction for generic (noni.i.d.) countable and continuous model classes. Section 3 critically reviews the classical principles (indifference, symmetry, minimax) for obtaining objective priors, introduces the universal prior inspired by Occam’s razor and quantified in terms of Kolmogorov complexity. In Section 4 (for i.i.d. ) and Section 5 (for universal ) we show various desirable properties of the universal prior and class (nonzero p(oste)rior, confirmation of universal hypotheses, reparametrization and regrouping invariance, no oldevidence and updating problem) in contrast to (most) classical continuous prior densities. Finally, we show that the universal mixture performs better than classical continuous mixtures, even in uncomputable environments. Section 6 contains critique and summary.
2 Bayesian Sequence Prediction
Notation. We use letters for natural numbers, and denote the cardinality of a set by or . We write for the set of finite strings over some alphabet , and for the set of infinite sequences. For a string of length we write with , and further abbreviate and . We assume that sequence is sampled from the “true” probability measure , i.e. is the probability that starts with . We denote expectations w.r.t. by . In particular for a function , we have . If is unknown but known to belong to a countable class of environments=models=measures , and forms a mutually exclusive and complete class of hypotheses, and is our prior belief in , then must be our (prior) belief in , and be our posterior belief in by Bayes’ rule. For a sequence of random variables, implies with probability 1 (w.p.1). Convergence is rapid in the sense that the probability that exceeds at more than times is bounded by . We sometimes loosely call this the number of errors.
Sequence prediction. Given a sequence , we want to predict its likely continuation . We assume that the strings which have to be continued are drawn from a “true” probability distribution . The maximal prior information a prediction algorithm can possess is the exact knowledge of , but often the true distribution is unknown. Instead, prediction is based on a guess of . While we require to be a measure, we allow to be a semimeasure [LV97, Hut04]:^{1}^{1}1Readers unfamiliar or uneasy with semimeasures can without loss ignore this technicality. Formally, is a semimeasure if , and a (probability) measure if equality holds and , where is the empty string. denotes the probability that a sequence starts with string . Further, is the “posterior” or “predictive” probability that the next symbol is , given sequence .
Bayes mixture. We may know or assume that belongs to some countable class of semimeasures. Then we can use the weighted average on (Bayesmixture, data evidence, marginal)
(1) 
for prediction. The most important property of semimeasure is its dominance
(2) 
which is a strong form of absolute continuity.
Convergence for deterministic environments. In the predictive setting we are not interested in identifying the true environment, but to predict the next symbol well. Let us consider deterministic first. An environment is called deterministic if for some sequence , and elsewhere (offsequence). In this case we identify with and the following holds:
(3) 
where is the weight of . This shows that rapidly converges to 1 and hence also for , and that is also a good multistep lookahead predictor. Proof: , since is monotone decreasing in and . Hence for any limit sequence . The bound follows from and .
Convergence in probabilistic environments. In the general probabilistic case we want to know how close . One can show that is to the true probability
(4) 
where can be the squared Euclidian or Hellinger or absolute or KL distance between and , or the squared Bayesregret [Hut04]. The first inequality actually holds for any two (semi)measures, and the last inequality follows from (2). These bounds (with ) imply
One can also show multistep lookahead convergence , (even for unbounded horizon ) which is interesting for delayed sequence prediction and in reactive environments [Hut04].
Continuous environmental classes. The bounds above remain approximately valid for most parametric model classes. Let be a family of probability distributions parameterized by a dimensional continuous parameter , and the true generating distribution. For a continuous weight density^{2}^{2}2 will always denote densities, and probabilities. the sums (1) are naturally replaced by integrals:
(5) 
The most important property of was the dominance (2) achieved by dropping the sum over . The analogous construction here is to restrict the integral over to a small vicinity of . Since a continuous parameter can typically be estimated to accuracy after observations, the largest volume in which as a function of is approximately flat is , hence . Under some weak regularity conditions one can prove [CB90, Hut04]
(6) 
where is the weight density (5) of in , and tends to zero for , and the average Fisher information matrix measures the local smoothness of and is bounded for many reasonable classes, including all stationary (order) finitestate Markov processes. We see that in the continuous case, is no longer bounded by a constant, but grows very slowly (logarithmically) with , which still implies that deviations are exponentially seldom. Hence, (6) allows to bound (4) even in case of continuous .
3 How to Choose the Prior
Classical principles. The probability axioms (implying Bayes’ rule) allow to compute posteriors and predictive distributions from prior ones, but are mute about how to choose the prior. Much has been written on the choice of noninformative=neutral=objective priors (see [KW96] for a survey and references; in Section 6 we briefly discuss how to incorporate subjective prior knowledge). For finite , Laplace’s symmetry or indifference argument which sets is a reasonable principle. The analogue uniform density for a compact measurable parameter space is less convincing, since becomes nonuniform under different parametrization (e.g. ). Jeffreys’ solution is to find a symmetry group of the problem (like permutations for finite or translations for ) and require the prior to be invariant under group transformations. Another solution is the minimax approach by Bernardo [CB90] which minimizes (the quite tight) bound (6) for the worst . Choice equalizes and hence minimizes (6). Problems are that there may be no obvious symmetry, the resulting prior can be improper, depend on which parameters are treated as nuisance parameters, on the model class, and on . Other principles are maximum entropy and conjugate priors. The principles above, although not unproblematic, can provide good objective priors in many cases of small discrete or compact spaces, but we will meet some more problems later. For “large” model classes we are interested in, i.e. countably infinite, noncompact, or nonparametric spaces, the principles typically do not apply or break down.
Occam’s razor et al. Machine learning, the computer science branch of statistics, often deals with very large model classes. Naturally, machine learning has (re)discovered and exploited quite different principles for choosing priors, appropriate for this situation. The overarching principles put together by Solomonoff [Sol64] are: Occam’s razor (choose the simplest model consistent with the data), Epicurus’ principle of multiple explanations (keep all explanations consistent with the data), (Universal) Turing machines (to compute, quantify and assign codes to all quantities of interest), and Kolmogorov complexity (to define what simplicity/complexity means).
We will first “derive” the so called universal prior, and subsequently justify it by presenting various welcome theoretical properties and by examples. The idea is that a priori, i.e. before seeing the data, all models are “consistent,” so apriori Epicurus would regard all models (in ) possible, i.e. choose . In order to also do (some) justice to Occam’s razor we should prefer simple hypothesis, i.e. assign high prior/low prior to simple/complex hypotheses . Before we can define this prior, we need to quantify the notion of complexity.
Notation. A function is said to be lower semicomputable (or enumerable) if the set is recursively enumerable. is upper semicomputable (or coenumerable) if is enumerable. is computable (or recursive) if and are enumerable. The set of (co)enumerable functions is recursively enumerable. We write for a constant of reasonable size: is reasonable, maybe even , but is not. We write for and
Kolmogorov complexity. We can now quantify the complexity of a string. Intuitively, a string is simple if it can be described in a few words, like “the string of one million ones”, and is complex if there is no such short description, like for a random object whose shortest description is specifying it bit by bit. We are interested in effective descriptions, and hence restrict decoders to be Turing machines (TMs). Let us choose some universal (socalled prefix) Turing machine with binary input=program tape, ary output tape, and bidirectional work tape. We can then define the prefix Kolmogorov complexity [LV97] of string as the length of the shortest binary program for which outputs :
For nonstring objects (like numbers and functions) we define , where is some standard code for . In particular, if is an enumeration of all (co)enumerable functions, we define .
An important property of is that it is nearly independent of the choice of . More precisely, if we switch from one universal TM to another, changes at most by an additive constant independent of . For reasonable universal TMs, the compiler constant is of reasonable size . A defining property of is that it additively dominates all coenumerable functions that satisfy Kraft’s inequality , i.e. for . The universal TM provides a shorter prefix code than any other effective prefix code. shares many properties with Shannon’s entropy (information measure) , but is superior to in many respects. To be brief, is an excellent universal complexity measure, suitable for quantifying Occam’s razor. We need the following properties of :

is not computable, but only upper semicomputable,

the upper bound , (7)

Kraft’s inequality , which implies for most ,

information nonincrease

if is enumerable and ,

is recursive and . if
Proofs of can be found in [LV97], and the (easy) proof of in the extended version of this paper.
The universal prior. We can now quantify a prior biased towards simple models. First, we quantify the complexity of an environment or hypothesis by its Kolmogorov complexity . The universal prior should be a decreasing function in the model’s complexity, and of course sum to (less than) one. Since satisfies Kraft’s inequality (3), this suggests the following choice:
(8) 
For this choice, the bound (4) on reads
(9) 
i.e. the number of times, deviates from by more than is bounded by , i.e. is proportional to the complexity of the environment. Could other choices for lead to better bounds? The answer is essentially no [Hut04]: Consider any other reasonable prior , where reasonable means (lower semi)computable with a program of size . Then, MDL bound (3) with and shows
Even for continuous classes , we can assign a (proper) universal prior (not density) for computable , and 0 for uncomputable ones. This effectively reduces to a discrete class which is typically dense in . We will see that this prior has many advantages over the classical prior densities.
4 Independent Identically Distributed Data
Laplace’s rule for Bernoulli sequences. Let be generated by a biased coin with head=1 probability , i.e. the likelihood of under hypothesis is , where . Bayes assumed a uniform prior density . The evidence is and the posterior probability weight density of after seeing is strongly peaked around the frequency estimate for large . Laplace asked for the predictive probability of observing after having seen , which is years = days since creation, so he concluded that the probability of doom, i.e. that the sun won’t rise tomorrow is .) This looks like a reasonable estimate, since it is close to the relative frequency, asymptotically consistent, symmetric, even defined for , and not overconfident (never assigns probability 1). . (Laplace believed that the sun had risen for
The problem of zero prior. But also Laplace’s rule is not without problems. The appropriateness of the uniform prior has been questioned in Section 3 and will be detailed below. Here we discuss a version of the zero prior problem. If the prior is zero, then the posterior is necessarily also zero. The above example seems unproblematic, since the prior and posterior densities and are nonzero. Nevertheless it is problematic e.g. in the context of scientific confirmation theory [Ear93].
Consider the hypothesis that all balls in some urn, or all ravens, are black (=1). A natural model is to assume that balls/ravens are drawn randomly from an infinite population with fraction of black balls/ravens and to assume a uniform prior over , i.e. just the BayesLaplace model. Now we draw objects and observe that they are all black.
We may formalize as the hypothesis . Although the posterior probability of the relaxed hypothesis , tends to 1 for for every fixed , remains identically zero, i.e. no amount of evidence can confirm . The reason is simply that zero prior implies zero posterior.
Note that refers to the unobservable quantity and only demands blackness with probability 1. So maybe a better formalization of is purely in terms of observational quantities: . Since , the predictive probability of observing further black objects is
One may speculate that the crux is the infinite population. But for a finite population of size and sampling with (similarly without) repetition, is close to one only if a large fraction of objects has been observed. This contradicts scientific practice: Although only a tiny fraction of all existing ravens have been observed, we regard this as sufficient evidence for believing strongly in .
There are two solutions of this problem: We may abandon strict/logical/allquantified/universal hypotheses altogether in favor of soft hypotheses like . Although not unreasonable, this approach is unattractive for several reasons. The other solution is to assign a nonzero prior to . Consider, for instance, the improper density , where is the Diracdelta (), or equivalently . We get is Kronecker’s . In particular , i.e. gets strongly confirmed by observing a reasonable number of black objects. This correct asymptotics also follows from the general result (3). Confirmation of is also reflected in the fact that tends much faster to zero than for uniform prior, i.e. the confidence that the next object is black is much higher. The power actually depends on the shape of around . Similarly gets confirmed: . On the other hand, if a single (or more) 0 are observed (), then the predictive distribution and posterior are the same as for uniform prior. , we get is much larger than for uniform prior. Since , where
The findings above remain qualitatively valid for i.i.d. processes over finite nonbinary alphabet and for nonuniform prior.
Surely to get a generally working setup, we should also assign a nonzero prior to and to all other “special” , like and , which may naturally appear in a hypothesis, like “is the coin or die fair”. The natural continuation of this thought is to assign nonzero prior to all computable . This is another motivation for the universal prior (8) constructed in Section 3. It is difficult but not impossible to operate with such a prior [PH04]. One may want to mix the discrete prior with a continuous (e.g. uniform) prior density, so that the set of noncomputable keeps a nonzero density. Although possible, we will see that this is actually not necessary.
Reparametrization invariance. Naively, the uniform prior is justified by the indifference principle, but as discussed in Section 3, uniformity is not reparametrization invariant. For instance if in our Bernoulli example we introduce a new parametrization , then the density is no longer uniform if is uniform.
More generally, assume we have some principle which leads to some prior . Now we apply the principle to a different parametrization and get prior . Assume that and are related via bijection . Another way to get a prior is to transform the prior . The reparametrization invariance principle (RIP) states that should be equal to .
For discrete , simply , and a uniform prior remains uniform () in any parametrization, i.e. the indifference principle satisfies RIP in finite model classes.
In case of densities, we have , and the indifference principle violates RIP for nonlinear transformations . But Jeffrey’s and Bernardo’s principle satisfy RIP. For instance, in the Bernoulli case we have and . , hence
Does the universal prior satisfy RIP? If we apply the “universality principle” to a parametrization, we get . On the other hand, simply transforms to ( is a discrete (nondensity) prior, which is nonzero on a discrete subset of ). For computable we have
Regrouping invariance. There are important transformations which are not bijections, which we consider in the following. A simple nonbijection is if we consider . More interesting is the following example: Assume we had decided not to record blackness versus nonblackness of objects, but their “color”. For simplicity of exposition assume we record only whether an object is black or white or colored, i.e. . In analogy to the binary case we use the indifference principle to assign a uniform prior on , where , and . All inferences regarding blackness (predictive and posterior) are identical to the binomial model with and or and and . Unfortunately, for uniform prior , is not uniform, i.e. the indifference principle is not invariant under splitting/grouping, or general regrouping. Regrouping invariance is regarded as a very important and desirable property [Wal96].
We now consider general i.i.d. processes . Dirichlet priors form a natural conjugate class () and are the default priors for multinomial (i.i.d.) processes over finite alphabet of size . Note that generalizes Laplace’s rule and coincides with Carnap’s [Ear93] confirmation function. Symmetry demands ; for instance for uniform and for BernardJeffrey’s prior. Grouping two “colors” and results in a Dirichlet prior with for the group. The only way to respect symmetry under all possible groupings is to set . This is Haldane’s improper prior, which results in unacceptably overconfident predictions . Walley [Wal96] solves the problem that there is no single acceptable prior density by considering sets of priors.
We now show that the universal prior is invariant under regrouping, and more generally under all simple (computable with complexity O(1)) even nonbijective transformations. Consider prior . If then transforms to (note that for nonbijections there is more than one consistent with ). In parametrization, the universal prior reads . Using (3) with and we get the universal prior is general transformation and hence regrouping invariant (within a multiplicative constant) w.r.t. simple computable transformations . , i.e.
Note that reparametrization and regrouping invariance hold for arbitrary classes and are not limited to the i.i.d. case.
5 Universal Sequence Prediction
Universal choice of . The bounds of Section 2 apply if contains the true environment . The larger the less restrictive is this assumption. The class of all computable distributions, although only countable, is pretty large from a practical point of view, since it includes for instance all of today’s valid physics theories. It is the largest class, relevant from a computational point of view. Solomonoff [Sol64, Eq.(13)] defined and studied the mixture over this class.
One problem is that this class is not enumerable, since the class of computable functions is not enumerable (halting problem), nor is it decidable whether a function is a measure. Hence is completely incomputable. Levin [ZL70] had the idea to “slightly” extend the class and include also lower semicomputable semimeasures. One can show that this class is enumerable, hence
(10) 
is itself lower semicomputable, i.e. , which is a convenient property in itself. Note that since
In some sense is the largest class of environments for which is in some sense computable [Hut04], but see [Sch02] for even larger classes.
The problem of old evidence. An important problem in Bayesian inference in general and (Bayesian) confirmation theory [Ear93] in particular is how to deal with ‘old evidence’ or equivalently with ‘new theories’. How shall a Bayesian treat the case when some evidence (e.g. Mercury’s perihelion advance) is known wellbefore the correct hypothesis/theory/model (Einstein’s general relativity theory) is found? How shall be added to the Bayesian machinery a posteriori? What is the prior of ? Should it be the belief in in a hypothetical counterfactual world in which is not known? Can old evidence confirm ? After all, could simply be constructed/biased/fitted towards “explaining” .
The universal class and universal prior formally solve this problem: The universal prior of is . This is independent of and of whether is known or not. If we use to construct or fit to explain , this will lead to a theory which is more complex () than a theory from scratch (), so cheats are automatically penalized. There is no problem of adding hypotheses to a posteriori. Priors of old hypotheses are not affected. Finally, includes all hypothesis (including yet unknown or unnamed ones) a priori. So at least theoretically, updating is unnecessary.
Other representations of . There is a much more elegant representation of : Solomonoff [Sol64, Eq.(7)] defined the universal prior as the probability that the output of a universal Turing machine starts with when provided with fair coin flips on the input tape. Note that a uniform distribution is also used in the socalled NoFreeLunch theorems to prove the impossibility of universal learners, but in our case the uniform distribution is piped through a universal Turing machine which defeats these negative implications. Formally, can be defined as
(11) 
where the sum is over all (socalled minimal) programs for which outputs a string starting with . may be regarded as a weighted mixture over all computable deterministic environments ( if and 0 else). Now, as a positive surprise, coincides with within an irrelevant multiplicative constant. So it is actually sufficient to consider the class of deterministic semimeasures. The reason is that the probabilistic semimeasures are in the convex hull of the deterministic ones, and so need not be taken extra into account in the mixture.
Bounds for computable environments. The bound (9) surely is applicable for and now holds for any computable measure . Within an additive constant the bound is also valid for and are excellent predictors with the only condition that the sequence is drawn from any computable probability distribution. Bound (9) shows that the total number of prediction errors is small. Similarly to (3) one can show that , where the monotone complexity is defined as the length of the shortest (nonhalting) program computing a string starting with [ZL70, LV97, Hut04]. . That is,
If is a computable sequence, then is finite, which implies on every computable sequence. This means that if the environment is a computable sequence (whichsoever, e.g. or the digits of or ), after having seen the first few digits, correctly predicts the next digit with high probability, i.e. it recognizes the structure of the sequence. In particular, observing an increasing number of black balls or black ravens or sunrises, () becomes rapidly confident that future balls and ravens are black and that the sun will rise tomorrow.
Universal is better than continuous . Although we argued that incomputable environments can safely be ignored, one may be nevertheless uneasy using Solomonoff’s 11) if outperformed by a continuous mixture (5) on such , for instance if would fail to predict a Bernoulli() sequence for incomputable . Luckily this is not the case: Although and can be incomputable, the studied classes themselves, i.e. the twoargument function , and the weight function , and hence , are typically computable (the integral can be approximated to arbitrary precision). Hence 10) and is often quite small. This implies for all by ( (
So any bound (6) for is directly valid also for , save an additive constant. That is, is superior (or equal) to all computable mixture predictors based on any (continuous or discrete) model class and weight , even if environment is not computable. Furthermore, while for essentially all parametric classes, grows logarithmically in for all (incl. computable) , is finite for computable . Bernardo’s prior even implies a bound for that is uniform (minimax) in . Many other priors based on reasonable principles (see Section 3 and [KW96]) and many other computable probabilistic predictors are argued for. The above actually shows that is superior to all of them.
6 Discussion
Critique and problems. In practice we often have extra information about the problem at hand, which could and should be used to guide the forecasting. One way is to explicate all our prior knowledge and place it on an extra input tape of our universal Turing machine , which leads to the conditional complexity . We now assign “subjective” prior to environment , which is large for those that are simple (have short description) relative to our background knowledge . Since
Another critique concerns the dependence of and on . Predictions for short sequences (shorter than typical compiler lengths) can be arbitrary. But taking into account our (whole) scientific prior knowledge , and predicting the now long string leads to good (less sensitive to “reasonable” ) predictions [Hut04].
Finally, and can serve as “gold standards” which practitioners should aim at, but since they are only semicomputable, they have to be (crudely) approximated in practice. Levin complexity [LV97], Schmidhuber’s speed prior, the minimal message and description length principles [Wal05], and offtheshelf compressors like LempelZiv are such approximations, which have been successfully applied to a plethora of problems [CV05, Sch04].
Summary. We compared traditional Bayesian sequence prediction based on continuous classes and prior densities to Solomonoff’s universal predictor , prior , and class . We discussed: Convergence for generic class and prior, the relative entropy bound for continuous classes, indifference/symmetry principles, the problem of zero p(oste)rior and confirmation of universal hypotheses, reparametrization and regrouping invariance, the problem of old evidence and updating, that works even in noncomputable environments, how to incorporate prior knowledge, the prediction of short sequences, the constant fudges in all results and the dependence, ’s incomputability and crude but practical approximations. In short, universal prediction solves or avoids or meliorates many foundational and philosophical problems, but has to be compromised in practice.
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