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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.

Table of Content

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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ID3 Algorithm and Hypothesis space in Decision Tree Learning

The collection of potential decision trees is the hypothesis space searched by ID3. ID3 searches this hypothesis space in a hill-climbing fashion, starting with the empty tree and moving on to increasingly detailed hypotheses in pursuit of a decision tree that properly classifies the training data.

In this blog, we’ll have a look at the Hypothesis space in Decision Trees and the ID3 Algorithm. 

ID3 Algorithm: 

The ID3 algorithm (Iterative Dichotomiser 3) is a classification technique that uses a greedy approach to create a decision tree by picking the optimal attribute that delivers the most Information Gain (IG) or the lowest Entropy (H).

What is Information Gain and Entropy?  

Information gain: .

The assessment of changes in entropy after segmenting a dataset based on a characteristic is known as information gain.

It establishes how much information a feature provides about a class.

We divided the node and built the decision tree based on the value of information gained.

The greatest information gain node/attribute is split first in a decision tree method, which always strives to maximize the value of information gain. 

The formula for Information Gain: 

Entropy is a metric for determining the degree of impurity in a particular property. It denotes the unpredictability of data. The following formula may be used to compute entropy:

S stands for “total number of samples.”

P(yes) denotes the likelihood of a yes answer.

P(no) denotes the likelihood of a negative outcome.

• Calculate the dataset’s entropy.
• For each feature/attribute.

Determine the entropy for each of the category values.

Calculate the feature’s information gain.

• Find the feature that provides the most information.
• Repeat it till we get the tree we want.

Characteristics of ID3: 

• ID3 takes a greedy approach, which means it might become caught in local optimums and hence cannot guarantee an optimal result.
• ID3 has the potential to overfit the training data (to avoid overfitting, smaller decision trees should be preferred over larger ones).
• This method creates tiny trees most of the time, however, it does not always yield the shortest tree feasible.
• On continuous data, ID3 is not easy to use (if the values of any given attribute are continuous, then there are many more places to split the data on this attribute, and searching for the best value to split by takes a lot of time).

Over Fitting:  

Good generalization is the desired property in our decision trees (and, indeed, in all classification problems), as we noted before. 

This implies we want the model fit on the labeled training data to generate predictions that are as accurate as they are on new, unseen observations.

Capabilities and Limitations of ID3:

• In relation to the given characteristics, ID3’s hypothesis space for all decision trees is a full set of finite discrete-valued functions.
• As it searches across the space of decision trees, ID3 keeps just one current hypothesis. This differs from the prior version space candidate Elimination approach, which keeps the set of all hypotheses compatible with the training instances provided.
• ID3 loses the capabilities that come with explicitly describing all consistent hypotheses by identifying only one hypothesis. It is unable to establish how many different decision trees are compatible with the supplied training data.
• One benefit of incorporating all of the instances’ statistical features (e.g., information gain) is that the final search is less vulnerable to faults in individual training examples.
• By altering its termination criterion to allow hypotheses that inadequately match the training data, ID3 may simply be modified to handle noisy training data.
• In its purest form, ID3 does not go backward in its search. It never goes back to evaluate a choice after it has chosen an attribute to test at a specific level in the tree. As a result, it is vulnerable to the standard dangers of hill-climbing search without backtracking, resulting in local optimum but not globally optimal solutions.
• At each stage of the search, ID3 uses all training instances to make statistically based judgments on how to refine its current hypothesis. This is in contrast to approaches that make incremental judgments based on individual training instances (e.g., FIND-S or CANDIDATE-ELIMINATION ).

Hypothesis Space Search by ID3: 

• ID3 climbs the hill of knowledge acquisition by searching the space of feasible decision trees.
• It looks for all finite discrete-valued functions in the whole space. Every function is represented by at least one tree.
• It only holds one theory (unlike Candidate-Elimination). It is unable to inform us how many more feasible options exist.
• It’s possible to get stranded in local optima.
• At each phase, all training examples are used. Errors have a lower impact on the outcome.

Searching Through the Hypothesis Space

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Part of the book series: Synthesis Lectures on Artificial Intelligence and Machine Learning ((SLAIML))

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“There are two possible outcomes: if the result confirms the hypothesis, then you’ve made a measurement. If the result is contrary to the hypothesis, then you’ve made a discovery.” — Enrico Fermi

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Settles, B. (2012). Searching Through the Hypothesis Space. In: Active Learning. Synthesis Lectures on Artificial Intelligence and Machine Learning. Springer, Cham. https://doi.org/10.1007/978-3-031-01560-1_3

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In Machine Learning , concept learning can be termed as “ a problem of searching through a predefined space of potential hypothesis for the hypothesis that best fits the training examples” – Tom Mitchell. In this article, we will go through one such concept learning algorithm known as the Find-S algorithm. If you want to go beyond this article and really want the level of expertise in you – you can get certified in Machine Learning Certification!

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The following topics are discussed in this article.

What is Find-S Algorithm in Machine Learning?

• How Does it Work?

Limitations of Find-S Algorithm

Implementation of find-s algorithm.

In order to understand Find-S algorithm, you need to have a basic idea of the following concepts as well:

• Concept Learning
• General Hypothesis
• Specific Hypothesis

1. Concept Learning 

Let’s try to understand concept learning with a real-life example. Most of human learning is based on past instances or experiences. For example, we are able to identify any type of vehicle based on a certain set of features like make, model, etc., that are defined over a large set of features.

These special features differentiate the set of cars, trucks, etc from the larger set of vehicles. These features that define the set of cars, trucks, etc are known as concepts.

Similar to this, machines can also learn from concepts to identify whether an object belongs to a specific category or not. Any algorithm that supports concept learning requires the following:

• Training Data
• Target Concept
• Actual Data Objects

2. General Hypothesis

Hypothesis, in general, is an explanation for something. The general hypothesis basically states the general relationship between the major variables. For example, a general hypothesis for ordering food would be  I want a burger.

G = { ‘?’, ‘?’, ‘?’, …..’?’}

3. Specific Hypothesis

The specific hypothesis fills in all the important details about the variables given in the general hypothesis. The more specific details into the example given above would be  I want a cheeseburger with a chicken pepperoni filling with a lot of lettuce. 

S = {‘Φ’,’Φ’,’Φ’, ……,’Φ’}

Now ,let’s talk about the Find-S Algorithm in Machine Learning.

The Find-S algorithm follows the steps written below:

• Initialize ‘h’ to the most specific hypothesis.
• The Find-S algorithm only considers the positive examples and eliminates negative examples. For each positive example, the algorithm checks for each attribute in the example. If the attribute value is the same as the hypothesis value, the algorithm moves on without any changes. But if the attribute value is different than the hypothesis value, the algorithm changes it to ‘?’.

Now that we are done with the basic explanation of the Find-S algorithm, let us take a look at how it works.

How Does It Work?

• The process starts with initializing ‘h’ with the most specific hypothesis, generally, it is the first positive example in the data set.
• We check for each positive example. If the example is negative, we will move on to the next example but if it is a positive example we will consider it for the next step.
• We will check if each attribute in the example is equal to the hypothesis value.
• If the value matches, then no changes are made.
• If the value does not match, the value is changed to ‘?’.
• We do this until we reach the last positive example in the data set.

There are a few limitations of the Find-S algorithm listed down below:

• There is no way to determine if the hypothesis is consistent throughout the data.
• Inconsistent training sets can actually mislead the Find-S algorithm, since it ignores the negative examples.
• Find-S algorithm does not provide a backtracking technique to determine the best possible changes that could be done to improve the resulting hypothesis.

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Now that we are aware of the limitations of the Find-S algorithm, let us take a look at a practical implementation of the Find-S Algorithm.

To understand the implementation, let us try to implement it to a smaller data set with a bunch of examples to decide if a person wants to go for a walk.

The concept of this particular problem will be on what days does a person likes to go on walk.

Looking at the data set, we have six attributes and a final attribute that defines the positive or negative example. In this case, yes is a positive example, which means the person will go for a walk.

So now, the general hypothesis is:

h 0 = {‘Morning’, ‘Sunny’, ‘Warm’, ‘Yes’, ‘Mild’, ‘Strong’}

This is our general hypothesis, and now we will consider each example one by one, but only the positive examples.

h 1 = {‘Morning’, ‘Sunny’, ‘?’, ‘Yes’, ‘?’, ‘?’}

h 2 = {‘?’, ‘Sunny’, ‘?’, ‘Yes’, ‘?’, ‘?’}

We replaced all the different values in the general hypothesis to get a resultant hypothesis. Now that we know how the Find-S algorithm works, let us take a look at an implementation using Python .

Let’s try to implement the above example using Python . The code to implement the Find-S algorithm using the above data is given below.

This brings us to the end of this article where we have learned the Find-S Algorithm in Mach ine Learning with its implementation and use case. I hope you are clear with all that has been shared with you in this tutorial.

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Statistics > Machine Learning

Title: hypothesis spaces for deep learning.

Abstract: This paper introduces a hypothesis space for deep learning that employs deep neural networks (DNNs). By treating a DNN as a function of two variables, the physical variable and parameter variable, we consider the primitive set of the DNNs for the parameter variable located in a set of the weight matrices and biases determined by a prescribed depth and widths of the DNNs. We then complete the linear span of the primitive DNN set in a weak* topology to construct a Banach space of functions of the physical variable. We prove that the Banach space so constructed is a reproducing kernel Banach space (RKBS) and construct its reproducing kernel. We investigate two learning models, regularized learning and minimum interpolation problem in the resulting RKBS, by establishing representer theorems for solutions of the learning models. The representer theorems unfold that solutions of these learning models can be expressed as linear combination of a finite number of kernel sessions determined by given data and the reproducing kernel.

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