Inference

Q: What is the difference between a hypothesis and a theory?
A: Think of a hypothesis as a card. A theory is a house made of hypotheses.
(From rec.humor.funny, attributed to Marilyn vos Savant.)

Introduction

People often distinguish between

• selecting a model class
• selecting a model from a given class and
• estimating the parameter values of a given model

A (parameter estimate of a) model (class) is not a prediction, at least not a prediction of future data, although it might be used to predict future data. It is an explanation of, a hypothesis about, the process that generated the given (past) data. It can be a good explanation or a bad explanation. Naturally we prefer good explanations.

Model Class

For example, polynomial models form a model class for sequences of points {<x_i,y_i> s.t. x_i < x_i+1}. The class includes constant functions, straight lines, quadratics, cubics, etc., and note that these have increasing numbers of parameters; they grow more complex. Note also that the class does not include functions based on trigonometric functions; they form another class.

Model

For example, the general cubic equation y = ax³+bx²+cx+d is a model for {<x_i,y_i>} s.t. x_i<x_i+1}. It has four parameters a, b, c & d which can be estimated to fit the model to some given data.

It is usually the case that a model has a fixed number of parameters (e.g., four above), but this can become blurred if hierarchical parameters or dependent parameters crop up.

Some writers reserve 'model' for a model (as above) where all the parameters have fixed, e.g., inferred, values; if there is any ambiguity I will (try to) use model instance, or fully parameterised model, for the latter. E.g., in these terms the normal distribution, N(μ,σ), is a model and N(0,1) is a model instance.

Parameter Estimation

For example, if we estimate the parameters of a cubic to be a=1, b=2, c=3 & d=4, we get the particular cubic polynomial y = x³+2x²+3x+4.

Hypothesis Complexity

Over-fitting

Over-fitting often appears as selecting a too complex model for the data. E.g., given ten data points from a physics experiment, a 9^th-degree polynomial could be fitted through them, exactly. This would almost certainly be a ridiculous thing to do. That small amount of data is probably better described by a straight line or by a quadratic with any minor variations explained as "noise" and experimental error.

Parameter estimation provides another manifestation of over-fitting: stating an estimated parameter value too precisely is also over-fitting.

• E.g., I toss a coin three times. It lands 'heads' once and 'tails' twice. What do we infer? p=0.5, p=0.3, p=0.33, p=0.333 or p=0.3333, etc.? In fact we learn almost nothing, except that the coin does have a head on one side and a tail on the other; p>0 and p<1.
• E.g., I toss a coin 30 times. It lands 'heads' 10 times and 'tails' 20 times. We are probably justified in starting to suspect a bias, perhaps 0.2<p<0.4.
• The precision with which the bias of a coin can be estimated can be made precise; see the [2-State / Binomial] distribution.

Classical statistics has developed a variety of significance tests to judge whether a model is justified by the data. An alternative is described below.

MML

Attempts to minimise the discrepancy between given data, D, and values implied by a hypothesis, H, almost always results in over-fitting, i.e. a too complex hypothesis (model, parameter estimate,...). E.g., if a quadratic gives a certain root mean squared (RMS) error, then a cubic will in general give a smaller RMS value.

Some penalty for the complexity of H is needed to give teeth to the so-called "law of diminishing returns". The minimum message length (MML) criterion is to consider a two-part message (remember [Bayes]):

2-part message: H; (D|H)

probabilities: P(H&D) = P(H).P(D|H)

message length: msgLen(H&D) = msgLen(H) + msgLen(D|H)

The first part of the two-part message can be considered to be a "header", as in data compression or data communication. Many file compression algorithms produce a header in the compressed file which states a number of parameter values etc., which are necessary for the data to be decoded.

The use of a prior, P(H), is considered to be controversial in classical statistics.

Notes

The idea of using compression to guide inference seems to have started in the 1960s.

• R. J. Solomonoff. A Formal Theory of Inductive Inference, I and II. Information and Control 7 pp.1-22 and pp.224-254, 1964
• A. N. Kolmogorov. Three approaches to the Quantitaive Definition of Information. Problems of Information and Transmission 1(1) pp.1-7, 1965
• G. J. Chaitin. On the Length of Programs for Computing Finite Binary Sequences. JACM 13(4) pp.547-569, Oct. 1966
• C. S. Wallace and D. M. Boulton. An Information Measure for Classification. CACM 11(2) pp.185-194, Aug. 1968
  This seems to be the first application of the principle to a real inference problem resulting in a practical and useful computer program.
  

More Formally

• Θ: variously the parameter-space, model class etc.
• θ or H: variously a particular parameter value, hypothesis, model, etc.
• P(θ) or P(H): prior probability of parameter value, hypothesis etc.
• X: data space, set of all possible observations
• D: data, an observation, D:X, often x:X
• P(D|H): the likelihood of the data D, is also written as f(D|H)
• P(H|D) or P(θ|D): the posterior probability of an estimate or hypothesis etc. given observed data
• P(D, H) = P(D & H): joint probability of D and H
• m(D) :X->Θ, estimator-function mapping observed data D onto an inferred parameter estimate, hypothesis, etc. as appropriate

Maximum Likelihood

The maximum likelihood principle is to choose H so as to maximise P(D|H).

E.g., Binomial Distribution (Bernouilli Trials)

A coin is tossed N times, landing heads, #head-times, and landing 'tails', #tail=N-#head times. We want to infer p=P(head). The likelihood of <#head,#tail> is:

P(#head|p) = p^#head.(1-p)^#tail

- log₂(P(#head|p)) = -#head.log₂(p) -#tail.log₂(1-p)

-#head/p + #tail/(1-p) = 0

#head/p = #tail/(1-p)

#head.(1-p) = (N-#head).p

#head = N.p

p = #head / N, q = 1-p = #tail / N

To sow some seeds of doubt, note that if the coin is thrown just once, the estimate for p must be either 0.0 or 1.0, which seems rather silly, although one could argue that such a small number of trials is itself rather silly. Still, if the coin were thrown 10 times and happened to land heads 10 times, which is conceivable, an estimate of 1.0 is still not sensible.

E.g., Normal Distribution

Given N data points, the maximum likelihood estimators for the parameters of a normal distribution, μ', σ' are given by:

μ' = {∑_i=1..N  x_i} / N   --the sample mean

σ'² = ∑_i=1..N  (x_i - μ')²} / N --the sample variance

Note that σ'² is biased, e.g., if there are just two data values it is implicitly assumed that they lie on opposite sides of the mean which plainly is not necessarily the case, i.e. the variance is under-estimated.

Notes

This section follows Farr (p.17..., 1999)

• G. Farr. Information Theory and MML Inference. School of Computer Science and Software Engineering, Monash University, 1999