Why the simplest explanation is always the best

Preamble

The simplest explanation is always the best among the explanations that are representative.  It is called the principle of parsimony,  Occam's razor. It is the bedrock of scientific enlightenment. There is a recent development that people start to do a category error to discard this principle based on the following setting that lead to misunderstanding: if we say that the simplest model that can explain the data in one representation but another representation requires a more complex model, then we are choosing more complex model. This is obviously wrong. The simplest explanation is to be chosen from the models that captures the complexity on the given representations, not the simplest over both representations. Filter explanations first based on representations then selection follows. Here we shown the core idea via an illustrative example. 

Figure: Circle has a zero Pearson
correlation. (Wikipedia)

Revisiting Occam’s Razor: A case of correlation and geometry

In order to understand this category error, we will work on a concrete example. Let's say we have a data, which is a circular shape $\mathscr{D}(x, y)$, as in figure. We have 3 models: 

$$\mathscr{M}_{1} : y = a x + b $$

$$\mathscr{M}_{2} : y = \sqrt{1-x^{2}} $$

$$\mathscr{M}_{3} : y ~ NN(x)  $$

And $\mathscr{U}$ a utility function, Pearson correlation $C(x, y)$.  We consider performance measure in ranking for the utility and a measure of representation. $\mathscr{M}_{3}$ is a neural network with a lot of parameters.

There is no error in choosing $\mathscr{M}_{1}$ based on the similar correlations these models produce. Principle of parsimony is not violated at all.  This is correct if we are only considering numerical representation. Pearson correlation as a utility works well for purely numerical representation. What about geometric representation? Then we need to change our utility function (representation measure or performance function). 

Let's say if we use curvature as a utility, $\kappa(x,y)$, in this case $\mathscr{M}_{1}$ fails to capture curvature and its is filtered out before Occam's razor can be applied. Then we left with  $\mathscr{M}_{2}$ and $\mathscr{M}_{3}$ . 

Correlation and geometric explanations are two different things. Two vastly different geometries can produce the same correlations.  A model can be quite good in explaining correlation but fails to capture geometric complexity. In this setting, it does not mean that Occam's razor is wrong. We need to apply Occam's razor across representations: numeric, geometric, algebraic, or symbolic, depending on the purpose. Always keep in mind the purpose or utility of model when invoking Occam's razor. Simplest explanation over the required representations that are relevant are the best. 

On the utility, performance and representations measure

Performance function, representation measure and utility functions can be different in real life, here for illustration purposes we consider them interchangeably. 

Conclusion

Nature minimises cost over complexity but under utility constraints. Minimal cost without satisfying utility or representation measure won't be chosen despite being the simplest. We need to filter first based on utility or representation measure before applying Occam's razor. The simplest explanation is always the best among the explanations that are representative. 

 Cite as 

 @misc{suzen25occam, 

     title = { Why the simplest explanation is always the best}, 

     howpublished = {\url{https://science-memo.blogspot.com/2025/11/simplest-explanation-always-best.html}}, 

     author = {Mehmet Süzen},

     year = {2025}

}  

Overfitting is about complexity ranking of inductive biases : Algorithmic recipe

Preamble

    Figure: Moon patterns
human brain
 invents. (Wikipedia)

Detecting overfitting is inherently a comparison problem of the complexity of multiple objects, i.e., models or an algorithm capable of making predictions. A model is overfitted (underfitted) if we only compare it to another model. Model selection involves comparing multiple models with different complexities. The summary of this approach with basic mathematical definitions is given here.

Misconceptions: Poor generalisation is not synonymous with overfitting. 

None of these techniques would prevent us from overfitting: Cross-validation, having more data, early stopping, and comparing test-train learning curves are all about generalisation. Their purpose is not to detect overfitting.

We need at least two different models, i.e., two different inductive biases, to judge which model is overfitted. One distinct approach in deep learning, called dropout, prevents overfitting while it alternates between multiple models, i.e., multiple inductive bias. For judgment, dropout implementation has to compare those alternating model test performances during training to judge overfitting. 

What is an inductive bias? 

There are multiple inceptions of inductive bias. Here, we concentrate on a parametrised model, $\mathscr{M}(\theta)$ on a dataset $\mathscr{D}$, the selection of a model type, or modelling approach, usually manifest as a functional form $\mathscr{M}=f(x)$ or as a function approximation, i.e., for example neural network, are all manifestation of inductive biases. Different parameterisation of model learned on the subsets of the dataset are still the same inductive bias.

Complexity ranking of inductive biases: An Algorithmic recipe 

We are sketching out an algorithmic recipe for complexity ranking of inductive biases via informal steps:

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 $argmin \{ \mathscr{C}_{1}, \mathscr{C}_{2}\}$

The core concept appears as when generalisations are close enough we pick out the inductive bias that is less complex. 

Conclusion & Outlook

In practice,  probably due to hectic delivery constraints, or mere laziness, we still rely on simple holdout method to build models, only single test and train split, not even learning curves, specially in deep learning models without practicing Occam's razor. A major insight in this direction appears to be that, holdout approach can only help us to detect generalisation, not overfitting. We clarify this via the concept of inductive bias distinguishing that different parametrisation of the same model doesn't change the inductive bias introduced by the modelling choice. 

In fact, due to resource constraints of model life-cycle, i.e., energy consumption and cognitive load of introducing a complex model, practicing proper Occam's razor: complexity ranking of inductive biases, is much more important than ever for sustainable environment and human capital.

Further reading

Some of the posts, reverse chronological order, that this blog have tried to convey what overfitting entails and its general implications. 

• Empirical risk minimization is not learning :A mathematical definition of learning and re-understanding of overfitting and Occam's razor in machine learning
• Critical look on why deployed machine learning model performance degrade quickly.
• Bringing back Occam's razor to modern connectionist machine learning.
• Understanding overfitting: an inaccurate meme in Machine Learning

Empirical risk minimization is not learning : A mathematical definition of learning and re-understanding of overfitting and Occam's razor in machine learning

    Simionescu Function (Wikipedia)

Preamble

The holy grail of machine learning appears to be the empirical risk minimisation. However, on the contrary to general dogma,  the primary objective of machine learning is not risk minimisation per se but mimicking human or animal learning. Empirical risk minimisation is just a snap-shot in this direction and is part of a learning measure, not the primary objective.

Unfortunately, all current major machine learning libraries are implementing empirical risk minimisation as primary objective, so called a training, manifest as usually .fit. Here we provide a mathematical definition of learning in the language of empirical risk minimisation and its implications on two very important concepts, overfitting and Occam's razor.

Our exposition is still informal but it should be readable for experienced practitioners.

Definition: Empirical Risk Minimization

Given set of $k$ observation $\mathscr{O} = \{o_{1}, ..., o_{k} \}$ where $o_{i} \in \mathbb{R}^{n}$, $n$-dimensional vectors.  Corresponding labels or binary classes, the set $\mathscr{S} = \{ s_{1}, .., s_{k}\}$, with $s_{i} \in \{0,1\}$ is defined. A function $g$  maps observations to classes $g: \mathscr{O} \to \mathscr{S}$.  An error function (or loss) $E$ measures the error made by the estimated map function $\hat{g}$ compare to true map function $g$,  $E=E(\hat{g}, g)$.  The entire idea of supervised machine learning boils down to minimising a functional called ER (Empirical Risk), here we denoted by $G$, it is a functional, meaning is a function of function, over the domain $\mathscr{D} = Tr(\mathscr{O} x \mathscr{S})$ in discrete form, $$ G[E] = \frac{1}{k} {\Large \Sigma}_{\mathscr{D} }  E(\hat{g}, g) $$.  This is so called a training a machine learning model, or an estimation for  $\hat{g}$. However, testing this estimate on the new data is not the main purpose of the learning.

Definition: Learning measure 

A learning measures $M$, on $\hat{g}$ is defined over set of $l$ observations with increasing size, $\Theta = \{ \mathscr{O}_{1}, ..., \mathscr{O}_{l}\}$ whereby size of each set is monotonically higher, meaning that $ | \mathscr{O}_{1}| < | \mathscr{O}_{2}| , ...,< | \mathscr{O}_{l}|$.

Definition: Empirical Risk Minimization with a learning measure (ERL)

Now, we are in a position to reformulate ER with learning measure, we call this ERL. This come with a testing procedure.

If empirical risks $G[E_{j}]$ lowers monotonically, $ G[E_{1}] > G[E_{2}] > ... > G[E_{l}]$, then we said the functional form of $\hat{g}$ is a learning over the set  $\Theta$.  

Functional form of $\hat{g}$ : Inductive bias

The functional form implies a model selection, and a technical term of this also known as inductive bias with other assumptions, meaning the selection of complexity of the model, for example a linear regression or nonlinear regression.

Re-understanding of overfitting and Occam's razor from ERL perspective 

If we have two different ERLs on $\hat{g}^{1}$ and $\hat{g}^{2}$. Then overfitting is a comparison problem between monotonically increasing empirical risks. If model, here an inductive bias or a functional form, over learning measure, we select the one with "higher monotonicity" and the less complex one and call the other overfitted model. Complexity here boils down to functional complexity of $\hat{g}^{1}$ and  $\hat{g}^{2}$  and overfitting can only be tested with two models over monotonicity (increasing) of ERLs.

Conclusions

In the age of deep learning systems, the classical learning theory needs an update on how do we define what is learning beyond a single shot fitting exercise. A first step in this direction would be to improve upon basic definitions of Empirical Risk (ER) minimisation that would reflect real-life learning systems similar to forgetting mechanism proposed by Ebbinghaus. This is consistent with Tom Mitchell's definition of operational machine learning. A next level would be to add causality in the definition.

Please cite as follows:

 @misc{suezen22erm, 

     title = { Empirical risk minimization is not learning : A mathematical definition of learning and re-understanding of overfitting and Occam's razor in machine learning}, 

     howpublished = {\url{http://science-memo.blogspot.com/2022/06/empirical-risk-minimisation-learning-curve.html}}, 

     author = {Mehmet Süzen},

     year = {2022}

}  

Postscript Notes

Following notes are added after initial release 

Postscript 1: Understanding overfitting as comparison of inductive biases

ERM could be confusing for even experienced researchers. It is indeed about risk measure. We measure the risk of a model, i.e., machine learning procedure that how much error would it make on the  given new data distribution, as in risk of investing. This is quite a similar notion as in financial risk of loss but not explicitly stated. 

Moreover, a primary objective of machine learning is not ERM but measure learning curves and pair-wise comparison of  inductive biases, avoiding overfitting.  An inductive bias, here we restrict the concept as in model  type,  is a model selection step: different  parametrisation of the same model are still the same inductive bias.  That’s why standard training-error learning curves can’t be used to detect overfitting alone. 

Postscript 2: Learning is not to optimise: Thermodynamic limit, true risk and accessible learning space

True risk minimisation in machine learning is not possible, instead we rely on ERM, i.e., Emprical Risk Minimisation.  However, the purpose of machine learning algorithm is not to minimise risk, as we only have a  partial knowledge about the reality through data.  Learning implies finding out a region  in accessible learning space whereby there is a monotonic increase in the objective; ERM is only a single point on this space, the concept rooted in German scientist Hermann Ebbinghaus  work on memory.

There is an intimate connection to thermodynamic limit and true risk in this direction as an open research.  However, it doesn’t imply infinite limit of data, but the observable’s behaviour. That’s why full empiricist approaches usually requires a complement of a physical laws,  such as Physics Informed Neural Networks (PINNs) or Structural Causal Model (SCM).

Postscript 3: Missing abstraction in modern machine learning libraries 

Interestingly current modern machine learning libraries stop abstracting further than fitting: .fit and .predict. This is short of learning as in machine learning. Learning manifest itself In learning curves. .learn functionality can be leveraged beyond fitting and if we are learning via monotonically increasing performance. Origin of this lack of tools for .learn appears to be how Empirical Risk Minimisation (ERM) is formulated on a single task.

Critical look on why deployed machine learning model performance degrade quickly

Illustration of William of Ockham 
(Wikipedia)

One of the major problems in using so called machine learning model, usually a supervised model, in so called deployment, meaning it will serve new data points which were not in the training or test set,  with great astonishment, modellers or data scientist observe that model's performance degrade quickly or it doesn't perform as good as test set performance. We earlier ruled out that underspecification would not be the main cause. Here we proposed that the primary reason of such performance degradation lies on the usage of hold out method in judging generalised performance solely.

Why model test performance does not reflect in deployment? Understanding overfitting

Major contributing factor is due to inaccurate meme of overfitting which actually meant overtraining and connecting overtraining erroneously to generalisation solely.  This was discussed earlier here as understanding overfitting. Overfitting is not about how good  is the function approximation compared to other subsets of the dataset of the same “model” works. Hence, the hold-out method (test/train) of measuring performances  does not  provide sufficient and necessary conditions to judge model’s generalisation ability: with this approach we can not detect overfitting (in Occam’s razor sense) and as well the deployment performance. 

How to mimic deployment performance?

This depends on the use case but the most promising approaches lies in adaptive analysis and detected distribution shifts and build models accordingly. However, the answer to this question is still an open research.