Saturday, 28 January 2023

Misconceptions on non-temporal learning: When do machine learning models qualify as prediction systems?

Preamble

    Babylonian Tablet for 
square root of 2.
 (Wikipedia)
Prediction implies a mechanics, as in knowing a form of a trajectory over time.  Strictly speaking a predictive system implies knowing a solution to the path, set of variable depending on time, time evolution of the system under consideration. Here, we define semi-informally how a prediction system is defined mathematically and show how non-temporal learning can be mapped into a prediction system. 

Temporal learning : Recurrence, trajectory and sequences

A trajectory can be seen as a function of time, identified in recurrence manner. It means $x(t_{i})=f(x_{i-1})$. However, this is one of the possible definitions. The physical equivalent of this appears as a solution to ordinary differential equation, such as the velocity $v(t) = dx(t)/dt$, recurrence on its solution. On the other hand machine learning, an empirical approach is taken and a sequence data such as natural language or a log events occurring in sequence. Any modelling on such data is called temporal learning. This includes classical time-series algorithms, gated units in deep learning and differential equations.

Definition: A prediction system  $\mathscr{F}$ that is build with data $D$ but utilised for a data that is not used in building it $D'$, qualified as such if both $D$ and $D'$ are temporal sets and output of the system is a horizon $\mathbb{H}$, that is a sequence. 

Using non-temporal supervised learning is interpolation or extrapolation

Often practice in industry to turn temporal interactions into flat set of data vectors,  $v_{i}$, $i$ corresponds to a time point or an arbitrary property of the dataset resulting in breaking the temporal associations and causal links.  This could also manifest as set of images with some labels which has no ordering or associational property in the dataset. Even though our system build upon these non-temporal datasets, indeed it constituted a learning systems as interpolation or extrapolation. Their utility in using them for $D'$, strictly speaking does not qualify as prediction systems. 

Mapping with pre-processing

A mapping indeed possible from non-temporal data to a temporal one, if their original form is not in temporal form yet. This is been studied in complexity literature. This requires an algorithm to map flattened data vectors we mentioned into a sequence data. 

Mapping with Causality

A distinct models from causal inference are qualified as predictive systems even if they are trained on non-temporal data, because causality establishes a temporal learning.

Non-temporal modals: Do they still learn?

Even though, we exclude non-temporal model utilisation as non-predictive systems, they still classified as learned models. Because their outputs are generated by a learning procedure. 

Conclusion

Differentiation among temporal and non-temporal learning is provided in associational manner. This results into definition of a prediction system, that excludes non-temporal machine learning models: such as models for unlinked set of vectors, a set of numbers mapped from any data modality. 

Further reading & postscript notes


Tuesday, 20 December 2022

The concept of overgeneralisation and goodness of rank : Overfitting is not about comparing training and test learning curves

Preamble 

    Walt Disney Hall,
Los Angeles  (Wikipedia)


Unfortunately, it is still thought in machine learning classes that overfitting can be detected by comparing training and test learning curves on the single model's performance.  The origins of this misconception is unknown. Looks like an urban legend has been diffused into main practice  and  even in academic works the misconception taken granted. Overfitting's definition appeared to be inherently about comparing complexities of two (or more) models. Models manifest themself as inductive biases modeller or data scientist inform in their tasks. This makes overfitting in reality a Bayesian concept at its core.  It is not about comparing training and test learning curves that if model is following a noise, but pairwise model comparison-testing procedure to select  more plausable belief among our beliefs that has the least information: entities should not be multiplied beyond necessity i.e., Occam's razor. We introduce a new concept in clarifying this practically, goodness of rank to distinguish from well known goodness of fit, and clarify concepts and provide steps to attribute models with overfitted or under-fitted models.

Poorly generalised model : Overgeneralisation or under-generalisation

The practice that is described in machine learning classes, and  practiced in industry that overfitting is about your model following training set closely but fails to generalised in test set. This is not overfitted model but a model that fails to generalise: a phenomena should be called Overgeneralisation (or under-generalisation). 

A procedure to detect overfitted model : Goodness of rank

We have provided complexity based abstract description of model selection procedure, here as complexity ranking: we will repeat this procedure with identification of the overfilled model explicitly.

In the following steps a sketch of an algorithmic recipe for complexity ranking of inductive biases via informal steps, overfitted model identification explicitly:

  1. Define a complexity measure $\mathscr{C}$($\mathscr{M}$) over an inductive bias.
  2. Define a generalisation measure  $\mathscr{G}$($\mathscr{M}$, $\mathscr{D}$) over and inductive bias and dataset.
  3. Select a set of inductive biases, at least-two, $\mathscr{M}_{1}$ and $\mathscr{M}_{2}$.
  4. Produce complexity and generalisation measures on ($\mathscr{M}$, $\mathscr{D}$): Here for two inductive biases: $\mathscr{C}_{1}$, $\mathscr{C}_{2}$,   $\mathscr{G}_{1}$, $\mathscr{G}_{2}$.
  5. Ranking of  $\mathscr{M}_{1}$ and $\mathscr{M}_{2}$:  $argmax \{ \mathscr{G}_{1}, \mathscr{G}_{2}\}$ and $argmin \{ \mathscr{C}_{1}, \mathscr{C}_{2}\}$ 
  6. $\mathscr{M}_{1}$ is an overfitted model compare to $\mathscr{M}_{2}$   if $\mathscr{G}_{1} <= \mathscr{G}_{2}$ and   $\mathscr{C}_{1} \gt \mathscr{C}_{2}$. 
  7. $\mathscr{M}_{2}$ is an overfitted model compare to $\mathscr{M}_{1}$ if $\mathscr{G}_{2} <= \mathscr{G}_{1}$ and   $\mathscr{C}_{2} \gt \mathscr{C}_{1}$.
  8. $\mathscr{M}_{1}$ is an underfitted model compare to $\mathscr{M}_{2}$   if $\mathscr{G}_{1} < \mathscr{G}_{2}$ and   $\mathscr{C}_{1} < \mathscr{C}_{2}$.
  9. $\mathscr{M}_{2}$ is an underfitted model compare to $\mathscr{M}_{1}$   if $\mathscr{G}_{2} < \mathscr{G}_{1}$ and   $\mathscr{C}_{2} < \mathscr{C}_{1}$.
If two model has the same complexity, then much better generalised model should be selected, in this case we can't conclude that either model is overfitted but generalised differently. Remembering that overfitting is about complexity ranking : Goodness of rank.

But overgeneralisation sounded like overfitting, isn't it?

Operationally overgeneralisation and overfitting implies two different things. Overgeneralisation operationally can be detected with a single model. Because, we can measure the generalisation performance of the model alone with data, in statistical literature this is called goodness of fit. Moreover overgeneralisation can also be called under-generalisation, as both implies poor generalisation performance.

However, overfitting implies a model overly performed compare to an other model i.e., model overfits but compare to what? Practically speaking, overgeneralisation can be detected via holdout method, but not overfitting. Overfitting goes beyond goodness of fit to goodness of rank as we provided recipe as pairwise model comparison.

Conclusion

The practice of comparing training and test learning curves for overfitting diffused into machine learning so deeply, the concept is almost always thought a bit in a fuzzy-way, even in distinguished lectures explicitly. Older textbooks and papers correctly identifies overfitting as comparison problem. As a practitioner, if we bear in mind that overfitting is about complexity ranking and it requires more than one model or inductive bias in order to be identified, then we are in better shape in selecting better model. Overfitting can not be detected via data alone on a single model.  


To make things clear, we provide concept definitions.

Generalisation A concept that if model can perform as good as the data it has not seen before, however seen here is a bit vague, it could have seen data points that are close to the data would be better suited in the context of supervised learning as oppose to compositional learning.

Goodness of fit An approach to check if model is generalised well.  

Goodness of rank An approach to check if model is overfitted or under-fitted comparable to other models.

Holdout method A metod to build a model on the portion of available data and measure the goodness of fit on the holdout part of the data, i.e., test and train.

Inductive bias  A set of assumptions data scientist made in building a representation of the real world, this manifest as model and the assumptions that come with a model.

Model  A model is a biased view of the reality from data scientist. Usually appears as a function of observables $X$ and parameters $\Theta$, $f(X, \Theta)$. The different values of $\Theta$ do not constitute a different model.  See also  What is a statistical model?,  Peter McCullagh 

Occam's razor (Principle of parsimony)  A principle that less complex explanation reflects reality better. Entities should not be multiplied beyond necessity.  

Overgeneralisation (Under-generalisation) If we have a good performance on the training set but very bad performance on the test set, model said to overgeneralise or under-generalise; as a result of goodness of fit testing, i.e., comparing learning curves over test and train datasets.

Regularisation An approach to augment model to improve generalisation.


Monday, 5 December 2022

The conditional query fallacy: Applying Bayesian inference from discrete mathematics perspective

Preamble

    The Tilled Field,
Joan Miró
(Wikipedia)
One of the core concepts in data sciences is conditional probabilities, $p(x|y)$ appear as logical description of many of the tasks, such as formulating regression or as a core concept in Bayesian Inference. However, there is operationally no special meaning of a conditional or joint probabilities as their arguments are no more than a compositional event statements. This raise a question: Is there any fundamental relationship between Bayesian Inference and discrete mathematics that is practically relevant to us as practitioners? Since, both topics are based on discrete statements returning a Boolean variables. Unfortunately, the answer to this question is a rabbit hole and probably even an open research.  There is no clearly established connections between discrete mathematics fundamentals and Bayesian Inference.  

Statement mappings as definition of probability 

Statement is a logical description of some events, or set of events. Let's have a semi-formal description of such statements.

Definition: A mathematical or logical statement formed with boolean relationships $\mathscr{R}$ (conjunctions) among set of events $\mathscr{E}$,  so a statement $\mathbb{S}$ is formed with at least a tuple of $\langle \mathscr{R},  \mathscr{E} \rangle$. 

Relationships can be any binary operator and events could explain anything perceptional, i.e., a discretised existence. This is the core discrete mathematics and almost all problems in this domain formed in this setting from defining functions to graph theory. A probability is no exception and definition naturally follows, as so called statement mapping.

Definition: A probability $\mathbb{P}$ is a statement mapping, $\mathbb{P}: \mathbb{S} \rightarrow [0,1]$. 

The interpretation of this definitions that a logical statement is always True if probability is 1 and always False if it is 0. However, having conditionals based on this is not that clear cut.

Conditional Query Fallacy 

A non-commutative statement may imply, reversing the order of statements should not yield to the same filtered set on the data for Bayesian Inference. However, Bayes' theorem would have a fallacy for statement mappings for conditionals in this sense. 

Definition: The conditional query fallacy is defined as one can not update belief in probability, because reversing order of statements in conditional probabilities halts Bayes' update,  i.e., back to back query results into the same dataset for inference.

At first glance, this appears as a Bayes' rule does not support commutative property, practically posterior being equal to likelihood.  However, this fallacy appears to be a notational misdirection. Inference on the filtered dataset back to back constituting a conditional fallacy i.e., when a query language is used to filter data to get A|B and B|A yielding to the same dataset regardless of filtering order. 

Although, in inference with data, likelihood is actually not a conditional probability, strictly speaking and not a filtering operation. It is merely a measure of update rule. We compute likelihood by multiplying values obtained by i.i.d. samples inserted into conjugate prior, a distribution is involved. Hence, the likelihood computationally is not really a reversal of conditional as in $P(A|B)$ written as reversed, $P(B|A)$.  

Outlook

In computing conditional probabilities for Bayesian Inference, our primary assumption is that conditional probabilities; likelihood and posterior are not identical. Discrete mathematics only allows Bayesian updates, if time evolution is explicitly stated with non-commutative statements for conditionals.

Going back to our initial question, indeed there is a deep connection between the fundamentals of discrete mathematics and Bayesian belief update on events as logical statements. The fallacy sounds a trivial error in judgement but (un)fortunately goes into philosophical definitions of probability that simultaneous tracking of time and sample space is not encoded in any of the notations explicitly, making statement filtering definition of probability a bit shaky.

Glossary of concepts

Statement Mapping A given set of mathematical statements mapped into a domain of numbers.

Probability A statement mapping, where domain is $\mathscr{D} = [0,1]$.

Conditional query fallacy Differently put than the above definition. Thinking that two conditional probabilities as reversed statements of each other in Bayesian inference, yields to the same dataset regardless of time-ordering of the queries.

Notes and further reading

Tuesday, 15 November 2022

Differentiating ensembles and sample spaces: Alignment between statistical mechanics and probability theory

Preamble 

Sample space is the primary concept introduced in any probability and statistics books and in papers. However, there needs to be more clarity about what constitutes a sample space in general: there is no explicit distinction between the unique event set and the replica sets. The resolution of this ambiguity lies in the concept of an ensemble.  The concept is first introduced by American theoretical physicist and engineer Gibbs in his book Elementary principle of statistical mechanics The primary utility of an ensemble is a mathematical construction that differentiates between samples and how they would form extended objects. 

In this direction, we provide the basics of constructing ensembles in a pedagogically accessible way from sample spaces that clears up a possible misconception. This usage of ensemble prevents the overuse of the term sample space for different things. We introduce some basic formal definitions.

    Figure: Gibbs's book
 introduced the concept of
ensemble (Wikipedia).

What Gibbs's had in mind by constructing statistical ensembles?

A statistical ensemble is a mathematical tool that connects statistical mechanics to thermodynamics. The concept lies in defining microscopic states for molecular dynamics; in statistics and probability, this corresponds to a set of events. Though these events are different at a microscopic level, they are sampled from a single thermodynamics ensemble, a representative of varying material properties or, in general, a set of independent random variables. In dynamics, micro-states samples an ensemble. This simple idea has helped Gibbs to build a mathematical formalism of statistical mechanics companion to Boltzmann's theories.

Differentiating sample space and ensemble in general

The primary confusion in probability theory on what constitutes a samples space is that there is no distinction between primitive events or events composed of primitive events. We call both sets sample space. This terminology easily overlooked in general as we concentrate on events set but not the primitive events set in solving practical problems.   

Definition: A primitive event $\mathscr{e}$ implies a logically distinct unit of experimental realisation that has not composed of any other events.

Definition: A sample space $\mathscr{S}$ is a set formed by all $N$ distinct primitive events $\mathscr{e}_{i}$.  

By this definition, regardless of how many fair coins are used or if a coin toss in a sequence for the experiment, the sample space is always ${H,T}$, because these are the most primitive distinct events a system can have, i.e., a single coin outcomes. However, the statistical ensemble can be different.  For example for two fair coins or coin toss in sequence of length two, corresponding ensemble of system size two reads ${HH, TT, HT, TH}$. Then, the definition of ensemble follows. 

Definition: An ensemble  $\mathscr{E}$ is a set of ordered set of primitive events $\mathscr{e}_{i}$. These event sets can be sampled with replacement but order matters, i.e., $ \{e_{i}, e_{j} \} \ne  \{e_{j}, e_{i} \}$, $i \ne j$.

Our two coin example's ensemble should be formally written as $\mathscr{E}=\{\{H,H\}, \{T,T\}, \{H,T\}, \{T,H\}\}$, as order matters members $HT$ and $TH$ are distinct. Obviously for a single toss ensemble and a sample space will be the same. 

Ergodicity makes the need for differentiation much more clear : Time and ensemble averaging 

The above distinction makes building time and ensemble averaging much easier. The term ensemble averaging is obvious as we know what would be the ensemble set and averaging over this set for a given observable.  Time averaging then could be achieved by curating a much larger set by resampling with replacement from the ensemble. Note that the resulting time-average value would not be unique, as one can generate many different sample sets from the ensemble. However, bear in mind that the definition of how to measure convergence to ergodic regime is not unique.

Conclusion

Even though the distinction we made sounds very obscure,  this alignment between statistical mechanics and probability theory may clarify the conception of ergodic regimes for general practitioners.

Further reading




(c) Copyright 2008-2020 Mehmet Suzen (suzen at acm dot org)

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