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Hypothesis in Machine Learning

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The concept of a hypothesis is fundamental in Machine Learning and data science endeavours. In the realm of machine learning, a hypothesis serves as an initial assumption made by data scientists and ML professionals when attempting to address a problem. Machine learning involves conducting experiments based on past experiences, and these hypotheses are crucial in formulating potential solutions.

It’s important to note that in machine learning discussions, the terms “hypothesis” and “model” are sometimes used interchangeably. However, a hypothesis represents an assumption, while a model is a mathematical representation employed to test that hypothesis. This section on “Hypothesis in Machine Learning” explores key aspects related to hypotheses in machine learning and their significance.

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How does a Hypothesis work?

Hypothesis space and representation in machine learning, hypothesis in statistics, faqs on hypothesis in machine learning.

A hypothesis in machine learning is the model’s presumption regarding the connection between the input features and the result. It is an illustration of the mapping function that the algorithm is attempting to discover using the training set. To minimize the discrepancy between the expected and actual outputs, the learning process involves modifying the weights that parameterize the hypothesis. The objective is to optimize the model’s parameters to achieve the best predictive performance on new, unseen data, and a cost function is used to assess the hypothesis’ accuracy.

In most supervised machine learning algorithms, our main goal is to find a possible hypothesis from the hypothesis space that could map out the inputs to the proper outputs. The following figure shows the common method to find out the possible hypothesis from the Hypothesis space:

Hypothesis Space (H)

Hypothesis space is the set of all the possible legal hypothesis. This is the set from which the machine learning algorithm would determine the best possible (only one) which would best describe the target function or the outputs.

Hypothesis (h)

A hypothesis is a function that best describes the target in supervised machine learning. The hypothesis that an algorithm would come up depends upon the data and also depends upon the restrictions and bias that we have imposed on the data.

The Hypothesis can be calculated as:

[Tex]y = mx + b [/Tex]

• m = slope of the lines
• b = intercept

To better understand the Hypothesis Space and Hypothesis consider the following coordinate that shows the distribution of some data:

Say suppose we have test data for which we have to determine the outputs or results. The test data is as shown below:

We can predict the outcomes by dividing the coordinate as shown below:

So the test data would yield the following result:

But note here that we could have divided the coordinate plane as:

The way in which the coordinate would be divided depends on the data, algorithm and constraints.

• All these legal possible ways in which we can divide the coordinate plane to predict the outcome of the test data composes of the Hypothesis Space.
• Each individual possible way is known as the hypothesis.

Hence, in this example the hypothesis space would be like:

The hypothesis space comprises all possible legal hypotheses that a machine learning algorithm can consider. Hypotheses are formulated based on various algorithms and techniques, including linear regression, decision trees, and neural networks. These hypotheses capture the mapping function transforming input data into predictions.

Hypothesis Formulation and Representation in Machine Learning

Hypotheses in machine learning are formulated based on various algorithms and techniques, each with its representation. For example:

• Linear Regression : [Tex] h(X) = \theta_0 + \theta_1 X_1 + \theta_2 X_2 + … + \theta_n X_n[/Tex]
• Decision Trees : [Tex]h(X) = \text{Tree}(X)[/Tex]
• Neural Networks : [Tex]h(X) = \text{NN}(X)[/Tex]

In the case of complex models like neural networks, the hypothesis may involve multiple layers of interconnected nodes, each performing a specific computation.

Hypothesis Evaluation:

The process of machine learning involves not only formulating hypotheses but also evaluating their performance. This evaluation is typically done using a loss function or an evaluation metric that quantifies the disparity between predicted outputs and ground truth labels. Common evaluation metrics include mean squared error (MSE), accuracy, precision, recall, F1-score, and others. By comparing the predictions of the hypothesis with the actual outcomes on a validation or test dataset, one can assess the effectiveness of the model.

Hypothesis Testing and Generalization:

Once a hypothesis is formulated and evaluated, the next step is to test its generalization capabilities. Generalization refers to the ability of a model to make accurate predictions on unseen data. A hypothesis that performs well on the training dataset but fails to generalize to new instances is said to suffer from overfitting. Conversely, a hypothesis that generalizes well to unseen data is deemed robust and reliable.

The process of hypothesis formulation, evaluation, testing, and generalization is often iterative in nature. It involves refining the hypothesis based on insights gained from model performance, feature importance, and domain knowledge. Techniques such as hyperparameter tuning, feature engineering, and model selection play a crucial role in this iterative refinement process.

In statistics , a hypothesis refers to a statement or assumption about a population parameter. It is a proposition or educated guess that helps guide statistical analyses. There are two types of hypotheses: the null hypothesis (H0) and the alternative hypothesis (H1 or Ha).

• Null Hypothesis(H 0 ): This hypothesis suggests that there is no significant difference or effect, and any observed results are due to chance. It often represents the status quo or a baseline assumption.
• Aternative Hypothesis(H 1 or H a ): This hypothesis contradicts the null hypothesis, proposing that there is a significant difference or effect in the population. It is what researchers aim to support with evidence.

Q. How does the training process use the hypothesis?

The learning algorithm uses the hypothesis as a guide to minimise the discrepancy between expected and actual outputs by adjusting its parameters during training.

Q. How is the hypothesis’s accuracy assessed?

Usually, a cost function that calculates the difference between expected and actual values is used to assess accuracy. Optimising the model to reduce this expense is the aim.

Q. What is Hypothesis testing?

Hypothesis testing is a statistical method for determining whether or not a hypothesis is correct. The hypothesis can be about two variables in a dataset, about an association between two groups, or about a situation.

Q. What distinguishes the null hypothesis from the alternative hypothesis in machine learning experiments?

The null hypothesis (H0) assumes no significant effect, while the alternative hypothesis (H1 or Ha) contradicts H0, suggesting a meaningful impact. Statistical testing is employed to decide between these hypotheses.

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Machine learning is a vast and complex field that has inherited many terms from other places all over the mathematical domain.

It can sometimes be challenging to get your head around all the different terminologies, never mind trying to understand how everything comes together.

In this blog post, we will focus on one particular concept: the hypothesis.

While you may think this is simple, there is a little caveat regarding machine learning.

The statistics side and the learning side.

Don’t worry; we’ll do a full breakdown below.

You’ll learn the following:

What Is a Hypothesis in Machine Learning?

• Is This any different than the hypothesis in statistics?
• What is the difference between the alternative hypothesis and the null?
• Why do we restrict hypothesis space in artificial intelligence?
• Example code performing hypothesis testing in machine learning

In machine learning, the term ‘hypothesis’ can refer to two things.

First, it can refer to the hypothesis space, the set of all possible training examples that could be used to predict or answer a new instance.

Second, it can refer to the traditional null and alternative hypotheses from statistics.

Since machine learning works so closely with statistics, 90% of the time, when someone is referencing the hypothesis, they’re referencing hypothesis tests from statistics.

Is This Any Different Than The Hypothesis In Statistics?

In statistics, the hypothesis is an assumption made about a population parameter.

The statistician’s goal is to prove it true or disprove it.

This will take the form of two different hypotheses, one called the null, and one called the alternative.

Usually, you’ll establish your null hypothesis as an assumption that it equals some value.

For example, in Welch’s T-Test Of Unequal Variance, our null hypothesis is that the two means we are testing (population parameter) are equal.

This means our null hypothesis is that the two population means are the same.

We run our statistical tests, and if our p-value is significant (very low), we reject the null hypothesis.

This would mean that their population means are unequal for the two samples you are testing.

Usually, statisticians will use the significance level of .05 (a 5% risk of being wrong) when deciding what to use as the p-value cut-off.

What Is The Difference Between The Alternative Hypothesis And The Null?

The null hypothesis is our default assumption, which we are trying to prove correct.

The alternate hypothesis is usually the opposite of our null and is much broader in scope.

For most statistical tests, the null and alternative hypotheses are already defined.

You are then just trying to find “significant” evidence we can use to reject our null hypothesis.

These two hypotheses are easy to spot by their specific notation. The null hypothesis is usually denoted by H₀, while H₁ denotes the alternative hypothesis.

Example Code Performing Hypothesis Testing In Machine Learning

Since there are many different hypothesis tests in machine learning and data science, we will focus on one of my favorites.

This test is Welch’s T-Test Of Unequal Variance, where we are trying to determine if the population means of these two samples are different.

There are a couple of assumptions for this test, but we will ignore those for now and show the code.

You can read more about this here in our other post, Welch’s T-Test of Unequal Variance .

We see that our p-value is very low, and we reject the null hypothesis.

What Is The Difference Between The Biased And Unbiased Hypothesis Spaces?

The difference between the Biased and Unbiased hypothesis space is the number of possible training examples your algorithm has to predict.

The unbiased space has all of them, and the biased space only has the training examples you’ve supplied.

Since neither of these is optimal (one is too small, one is much too big), your algorithm creates generalized rules (inductive learning) to be able to handle examples it hasn’t seen before.

Here’s an example of each:

Example of The Biased Hypothesis Space In Machine Learning

The Biased Hypothesis space in machine learning is a biased subspace where your algorithm does not consider all training examples to make predictions.

This is easiest to see with an example.

Let’s say you have the following data:

Happy  and  Sunny  and  Stomach Full  = True

Whenever your algorithm sees those three together in the biased hypothesis space, it’ll automatically default to true.

This means when your algorithm sees:

Sad  and  Sunny  And  Stomach Full  = False

It’ll automatically default to False since it didn’t appear in our subspace.

This is a greedy approach, but it has some practical applications.

Example of the Unbiased Hypothesis Space In Machine Learning

The unbiased hypothesis space is a space where all combinations are stored.

We can use re-use our example above:

This would start to breakdown as

Happy  = True

Happy  and  Sunny  = True

Happy  and  Stomach Full  = True

Let’s say you have four options for each of the three choices.

This would mean our subspace would need 2^12 instances (4096) just for our little three-word problem.

This is practically impossible; the space would become huge.

So while it would be highly accurate, this has no scalability.

More reading on this idea can be found in our post, Inductive Bias In Machine Learning .

Why Do We Restrict Hypothesis Space In Artificial Intelligence?

We have to restrict the hypothesis space in machine learning. Without any restrictions, our domain becomes much too large, and we lose any form of scalability.

This is why our algorithm creates rules to handle examples that are seen in production. 

This gives our algorithms a generalized approach that will be able to handle all new examples that are in the same format.

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Hypothesis Space

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• Hendrik Blockeel  

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Model space

The hypothesis space used by a machine learning system is the set of all hypotheses that might possibly be returned by it. It is typically defined by a Hypothesis Language , possibly in conjunction with a Language Bias .

Motivation and Background

Many machine learning algorithms rely on some kind of search procedure: given a set of observations and a space of all possible hypotheses that might be considered (the “hypothesis space”), they look in this space for those hypotheses that best fit the data (or are optimal with respect to some other quality criterion).

To describe the context of a learning system in more detail, we introduce the following terminology. The key terms have separate entries in this encyclopedia, and we refer to those entries for more detailed definitions.

A learner takes observations as inputs. The Observation Language is the language used to describe these observations.

The hypotheses that a learner may produce, will be formulated in...

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De Raedt, L. (1992). Interactive theory revision: An inductive logic programming approach . London: Academic Press.

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Nédellec, C., Adé, H., Bergadano, F., & Tausend, B. (1996). Declarative bias in ILP. In L. De Raedt (Ed.), Advances in inductive logic programming . Frontiers in artificial intelligence and applications (Vol. 32, pp. 82–103). Amsterdam: IOS Press.

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School of Computer Science and Engineering, University of New South Wales, Sydney, Australia, 2052

Claude Sammut

Faculty of Information Technology, Clayton School of Information Technology, Monash University, P.O. Box 63, Victoria, Australia, 3800

Geoffrey I. Webb

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Blockeel, H. (2011). Hypothesis Space. In: Sammut, C., Webb, G.I. (eds) Encyclopedia of Machine Learning. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-30164-8_373

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7.8.1 Version-Space Learning

Rather than enumerating all of the hypotheses, the set of elements of ℋ consistent with all of the examples can be found more efficiently by imposing some structure on the hypothesis space.

Hypothesis h 1 is a more general hypothesis than hypothesis h 2 if h 2 implies h 1 . In this case, h 2 is a more specific hypothesis than h 1 . Any hypothesis is both more general than itself and more specific than itself.

Example 7.29 .

The hypothesis ¬ ⁢ a ⁢ c ⁢ a ⁢ d ⁢ e ⁢ m ⁢ i ⁢ c ∧ m ⁢ u ⁢ s ⁢ i ⁢ c is more specific than m ⁢ u ⁢ s ⁢ i ⁢ c and is also more specific than ¬ ⁢ a ⁢ c ⁢ a ⁢ d ⁢ e ⁢ m ⁢ i ⁢ c . Thus, m ⁢ u ⁢ s ⁢ i ⁢ c is more general than ¬ ⁢ a ⁢ c ⁢ a ⁢ d ⁢ e ⁢ m ⁢ i ⁢ c ∧ m ⁢ u ⁢ s ⁢ i ⁢ c . The most general hypothesis is t ⁢ r ⁢ u ⁢ e . The most specific hypothesis is f ⁢ a ⁢ l ⁢ s ⁢ e .

The “more general than” relation forms a partial ordering over the hypothesis space. The version-space algorithm that follows exploits this partial ordering to search for hypotheses that are consistent with the training examples.

Given hypothesis space ℋ and examples E , the version space is the subset of ℋ that is consistent with the examples.

The general boundary of a version space, G , is the set of maximally general members of the version space (i.e., those members of the version space such that no other element of the version space is more general). The specific boundary of a version space, S , is the set of maximally specific members of the version space.

These concepts are useful because the general boundary and the specific boundary completely determine the version space, as shown by the following proposition.

Proposition 7.2 .

The version space is the set of h ∈ H such that h is more general than an element of S and more specific than an element of G .

Candidate Elimination Learner

The candidate elimination learner incrementally builds the version space given a hypothesis space ℋ and a set E of examples. The examples are added one by one; each example possibly shrinks the version space by removing the hypotheses that are inconsistent with the example. The candidate elimination algorithm does this by updating the general and/or the specific boundary for each new example. This is described in Figure 7.21 .

Example 7.30 .

Consider how the candidate elimination algorithm handles Example 7.26 , where H is the set of conjunctions of literals.

Before it has seen any examples, G 0 = { t ⁢ r ⁢ u ⁢ e } – the user reads everything – and S 0 = { f ⁢ a ⁢ l ⁢ s ⁢ e } – the user reads nothing. Note that t ⁢ r ⁢ u ⁢ e is the empty conjunction and f ⁢ a ⁢ l ⁢ s ⁢ e is the conjunction of an atom and its negation.

After considering the first example, a 1 , G 1 = { t ⁢ r ⁢ u ⁢ e } and

Thus, the most general hypothesis is that the user reads everything, and the most specific hypothesis is that the user only reads articles exactly like this one.

After considering the first two examples, G 2 = { t ⁢ r ⁢ u ⁢ e } and

Since a 1 and a 2 disagree on music, but have the same prediction it can be concluded that music cannot be relevant.

After considering the first three examples, the general boundary becomes

and S 3 = S 2 . Now there are two most general hypotheses; the first is that the user reads anything about crime, and the second is that the user reads anything non-academic.

After considering the first four examples,

and S 4 = S 3 .

After considering all five examples,

Thus, after five examples, only two hypotheses exist in the version space. They differ only in their prediction on an example that has c ⁢ r ⁢ i ⁢ m ⁢ e ∧ ⁢ l ⁢ o ⁢ c ⁢ a ⁢ l true. If the target concept can be represented as a conjunction, only an example with c ⁢ r ⁢ i ⁢ m ⁢ e ∧ ⁢ l ⁢ o ⁢ c ⁢ a ⁢ l true will change G or S . This version space can make predictions about all other examples.

The Bias Involved in Version-Space Learning

Recall that a bias is necessary for any learning to generalize beyond the training data. There must have been a bias in Example 7.30 because, after observing only five of the 16 possible assignments to the input variables, an agent was able to make predictions about examples it had not seen.

The bias involved in version-space learning is a called a language bias or a restriction bias because the bias is obtained from restricting the allowable hypotheses. For example, a new example with crime false and music true will be classified as false (the user will not read the article), even though no such example has been seen. The restriction that the hypothesis must be a conjunction of literals is enough to predict its value.

This bias should be contrasted with the bias involved in decision tree learning . A decision tree can represent any Boolean function. Decision tree learning involves a preference bias , in that some Boolean functions are preferred over others; those with smaller decision trees are preferred over those with larger decision trees. A decision tree learning algorithm that builds a single decision tree top-down also involves a search bias in that the decision tree returned depends on the search strategy used.

The candidate elimination algorithm is sometimes said to be an unbiased learning algorithm because the learning algorithm does not impose any bias beyond the language bias involved in choosing ℋ . It is easy for the version space to collapse to the empty set, for example, if the user reads an article with crime false and music true. This means that the target concept is not in ℋ . Version-space learning is not tolerant to noise; just one misclassified example can throw off the whole system.

The bias-free hypothesis space is where ℋ is the set of all Boolean functions. In this case, G always contains one concept: the concept which says that all negative examples have been seen and every other example is positive. Similarly, S contains the single concept which says that all unseen examples are negative. The version space is incapable of concluding anything about examples it has not seen; thus, it cannot generalize. Without a language bias or a preference bias, no generalization and, therefore, no learning will occur.

Artificial Intelligence: Foundations of Computational Agents, Poole & Mackworth This online version is free to view and download for personal use only. The text is not for re-distribution, re-sale or use in derivative works. Copyright © 2017, David L. Poole and Alan K. Mackworth . This book is published by Cambridge University Press .

Genetic algorithm: Hypothesis space search

As already understood from our illustrative example, it is clear that genetic algorithms employ a randomized beam search method to seek maximally fit hypotheses. In the hypothesis space search method, we can see that the gradient descent search in backpropagation moves smoothly from one hypothesis to another. On the other hand, the genetic algorithm search can move much more abruptly. It replaces the parent hypotheses with an offspring that can be very different from the parent. Due to this reason, genetic algorithm search has lower chances of it falling into the same kind of local minima that plaques the gradient descent methods.

There is one practical difficulty that is often encountered in genetic algorithms, it is crowding. Crowding can be defined as the phenomenon in which some individuals that are more fit in comparison to others, reproduce quickly, therefore the copies of this individual take over a larger fraction of the population. Most of the strategies used in the genetic algorithms are inspired by biological evolution. One such other strategy used is fitness sharing, in which the measured fitness of an individual is decreased by the presence of another individual of a similar kind. The third method is to restrict all the individuals to combine to form offspring. To better understand we can say that by allowing individuals of the same kind to recombine, clusters of similar individuals are formed, forming multiple subspecies in the population.

Another method would be to spatially distribute individuals and allow only nearby individuals to combine.

Population evolution and schema theorem.

The schema theorem of Holland is used to mathematically characterize the evolution over time of the population with respect to time. It is based on the concept of schema. So, what is schema? Schema is any string composed of 0s, and 1s, and *s, where * represents null, so a schema 0*10, is the same as 0010 and 0110. The schema theorem characterizes the evolution within a genetic algorithm on the basis of the number of instances representing each schema. Let us assume the m(s, t) to denote the number of instances of schema denoted by ‘s’, in the population at the time ‘t’, the expected value in the schema theorem is described as m(s, t+1), in terms of m(s, t), and the other parameters of the population, schema, and GA.

In a genetic algorithm, the evolution of the population depends on the selection step, the recombination step, and the mutation step. The schema theorem is one of the most widely used theorems in the characterization of population evolution within a genetic algorithm. If it fails to consider the positive effects of crossover and mutation, it is in a way incomplete. There are many other recent theoretical analyses that have been proposed, many of these analogies are based on models such as Markov chain models and the statistical mechanical model.

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Title: the platonic representation hypothesis.

Abstract: We argue that representations in AI models, particularly deep networks, are converging. First, we survey many examples of convergence in the literature: over time and across multiple domains, the ways by which different neural networks represent data are becoming more aligned. Next, we demonstrate convergence across data modalities: as vision models and language models get larger, they measure distance between datapoints in a more and more alike way. We hypothesize that this convergence is driving toward a shared statistical model of reality, akin to Plato's concept of an ideal reality. We term such a representation the platonic representation and discuss several possible selective pressures toward it. Finally, we discuss the implications of these trends, their limitations, and counterexamples to our analysis.

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