Likelihood vs Probability: What’s the Difference?

Introduction

Likelihood and probability are interrelated and often confused terms finding common usage in data science and business. Both probabilities are associated with probabilities but differ in definition and usage. The article aims to clarify likelihood vs probability definitions, usage, and misconceptions for better understanding and application in the respective field.

Likelihood vs Probability – Overview

What is Likelihood?

We can define likelihood as a quantitative estimation or measure that states the fitness of a model or hypothesis in observed data. It can also be interpreted as the chance of finding the desired result or data collection in a specific parameter set. Playing a fundamental role in statistical inference, the ultimate aim of likelihood is to conclude about the data’s characteristics. The role in achieving the same is seen through parameter estimation, which utilizes Maximum Likelihood Estimation or MLE to find parameter estimates.

Hypothesis testing uses likelihood ratios to assess the null hypothesis. Similarly, likelihood contributes by comparing models for model selection and checking. Researchers commonly utilize Bayesian Information Criterion (BIC) and Akaike Information Criterion (AIC) as measures in model selection. Likelihood-based methods play a significant role in constructing confidence intervals to estimate the parameters.

What is Probability?

Probability refers to the possibility or chances of occurrence of a specific outcome that we predict according to the model parameters. The probability measure provides a framework for prediction and understanding the possibility of uncertain events. It helps to quantify uncertainty for probability theory by comparison of the likelihood of different outcomes. In predictive modeling, we use probability theory to construct confidence intervals, make probabilistic predictions, and perform hypothesis testing.

Furthermore, the randomness and stochastic processes depend on probability theory due to the requirement to analyze and model random phenomena. Here probability is used for simulation and understanding of complex systems. Additionally, the axioms, rules, and theorems important for the analysis of uncertainty and logical consistency are provided through probability.

Examples of Probability and Likelihood

Examples 1 – Coin Toss

In the context of coin tosses, likelihood and probability represent different aspects of the same experiment. The likelihood refers to the probability of observing a specific outcome given a particular model or hypothesis. On the other hand, probability represents the long-term frequency of an event occurring over multiple trials.

Let’s consider a fair coin toss. The likelihood of obtaining a ‘heads’ outcome in a single toss, assuming the coin is fair, is 0.5 since there are two equally likely possibilities (heads or tails). However, if we have observed a sequence of coin tosses and obtained five ‘heads’ and five ‘tails,’ the likelihood of this specific sequence occurring with a fair coin is different from 0.5.

Now, let’s demonstrate the probabilities and likelihoods of different outcomes in a table:

In this example, the probability of each outcome (heads or tails) remains constant at 0.5 for a fair coin. However, the likelihood of specific outcomes changes based on the observed data, reflecting the uncertainty associated with the underlying model or hypothesis.

Example 2 – Spinners

Consider a spinner with four equal-sized sections: red, blue, green, and yellow. In the context of spinners, likelihood and probability also represent different aspects of the experiment.

The likelihood of landing on a specific color, given that the spinner is fair and unbiased, is 0.25 (1/4) for each color. This is because there are four equally likely possibilities.

On the other hand, the probability of landing on each color remains constant at 0.25 since the spinner’s sections are equally sized.

Let’s demonstrate the probabilities and likelihoods of different outcomes in a table:

In this example, both the probability and likelihood of each color landing remain fixed at 0.25, as the spinner’s sections are uniformly sized. The distinction between likelihood and probability becomes apparent when considering specific observed outcomes, as the likelihood can vary based on the data collected from multiple spins. Probability, however, remains constant for each possible color, reflecting the spinner’s unbiased nature.

Likelihood vs Probability – Calculation

Calculation of Likelihood Using the Likelihood Function

The likelihood function is a mathematical expression that helps identify the data distribution. The function is denoted as, Likelihood(|x), where stands for parameters of the desired model and X represents the observed data.

Let us understand this with an example. For instance, you have a bag of colored marbles. You want to predict the probability of picking a red marble. Begin with random draws, record the colors, and then calculate the likelihood using the stated formula. You will calculate or estimate the parameter representing the probability of drawing red marble. We will denote the likelihood function, as previously stated, which states the probability of observing a given data x for a specific value.

Assuming the independent and identically distributed draws, the likelihood function will be:

L(|x)=k(1-)(n-k), where n is the number of draws and k is the number of red marbles in observed data.

Let us assume you draw the marble five times in the sequence, red, red, blue, red, and blue.

• L(0.5|x)=0.53(1-0.5)(5-3)
• L(0.5|x)=0.530.52
• L(0.5|x)=0.015625

Thus, at = 0.5, the likelihood of observing the stated drawing of the stated sequence of balls is 0.015625.

Calculation of Probability Using PMF or PDF

The PMF calculates the probability of finding a desired value from a definite finite set of variables. It is expressed as

P(X=x), where the x is the particular value of a random variable

In PMF, the value of x is non-negative, and the sum of probabilities based on the possible values of x is 1.

The PDF covers a broad spectrum and indicates the probability of finding specific values or falling in a specific range of values. Here the expression is expressed as f(x). Again, the probability density function is non-negative, and the area covered by the curve = 1.

Interpretation of Likelihood as a Measure of How Well the Data Fits a Specific Hypothesis or Model

Keeping the values in the above-stated formula, the range of values will vary depending on the situation. But the higher likelihood value indicates a positive result and higher relatability between the observed and calculated values.

Examples of Likelihood and Probability

Examples Illustrating the Concept of Likelihood

Let us take the example of a coin toss. You have a fair coin that you toss around ten times. Now you need to assess the fairness or biasedness of the coin. You need to set up a parameter; let’s say the fairness hypothesis is that eight heads and two tails indicate the coin is fair. The high likelihood tends to represent the fair coin, further supporting the fairness hypothesis.

Taking another example of Gaussian distribution, assume a dataset of 100 measurements following the same. You want to know the mean and standard deviation of the distribution. The different combinations can be set based on the parameters where the high probability estimate will indicate the maximum likelihood for the best Gaussian distribution.

Examples Demonstrating the Concept of Probability

Let us understand the probability also with the coin toss example. You can get only two results on tossing the coin: head or tail. Hence, the probability of each is 0.5, and the sum of probabilities is 1. Thus, it states all possible outcomes.

Another example is a dice roll, where the dice are six-faced. The probability of obtaining a specific number on dice is 16, while the sum of probabilities will be (616)=1.

Likelihood vs. Probability in Data Science

Application of Likelihood in Data Science

We use Maximum Likelihood Estimation (MLE), also known as the likelihood function in parameter estimation, to find the value of parameters. The values are as per the maximum likelihood of observed data. In model selection, the likelihood compares the different models to find the best fit. Examples of techniques include the likelihood ratio test and Bayesian Information Criterion (BIC). Hypothesis testing checks the data based on different hypotheses. It also involves comparison however differs from the model selection.

Application of Probability in Data Science

• Predictive modeling in businesses relies on probability for logical predictions and future planning.
• Probability is used in classification tasks through different algorithms, such as Naive Bayes classifiers and logistic regression.
• Bayesian regression methods help model uncertainty in regression tasks by finding the posterior distribution of the target variable.
• Time series forecasting utilizes techniques like ARIMA and state space models to calculate uncertainty in future predictions.
• Anomaly detection uses probability distributions to identify deviations from expected behavior, employing techniques like Gaussian mixture models and Hidden Markov Models.
• Bayesian inference combines prior knowledge with observed data to generate posterior probabilities, aiding belief update and uncertainty improvement.
• Bayesian inference is used for hypothesis testing, model selection, and comparison.
• Uncertainty quantification involves assessing and quantifying predictions and model parameters using Monte Carlo simulations and bootstrapping methods.
• Reliability analysis assesses the probability of failure or success of a system or component.
• Design optimization incorporates uncertainty by considering probabilistic objectives and constraints, utilizing probability distributions and optimization algorithms.

Integration of Likelihood and Probability

Probabilistic graphical models are for the probability distribution of a set of random variables, while the likelihood is suitable for parameter estimation. For prediction-based analysis, it combines already available probabilities with probabilistic graphical models, such as the Bayesian framework. When coupled with likelihood, Bayesian learning updates the prior beliefs further, resulting in a new analysis combining prior and new beliefs. It leads to the application of likelihood vs probability in risk assessment.

The statistical learning methods include maximum likelihood estimation, neural networks, and support vector machines that primarily optimize objective functions using likelihood calculations. The combination serves the purpose of finding decision boundaries and the best model parameters.

Common Misconceptions and Pitfalls

1. Addressing Common Misunderstandings About Likelihood and Probability in the Context of Data Science

One of the common misunderstandings includes assuming both likelihood and probability to be the same thing. Rather, they are different concepts where likelihood mainly deals with model selection and parameter estimation. Alternatively, probability is more focused on uncertainty quantification and predictive modeling.

Another misunderstanding is assuming that likelihood represents the probability of a true hypothesis. The likelihood states the measure of the quality of how data fits the specific hypothesis or model. It is about the relation between parameters and observed data.

2. Clarifying Misconceptions Regarding Their Interchangeability and Appropriate Usage

Concerning interchangeability, people think both terms are interchangeable. But they are not. For instance, likelihood vs probability in risk assessment is the same yet different. The likelihood states the relation between parameter values and observed data. In contrast, the probability is the possibility of the occurrence of an event. Their usage is also different, where the likelihood is mainly for prediction and parameter estimation, but the probability is more suited for predicting future events.

Also Read: Statistics and Probability Concepts for Data Science

Conclusion

We hope you learnt all about likelihood vs probability with our article. Likelihood and probability are different concepts. Their usage and application also differ, along with the techniques used to find the specific results. The latter focuses on the occurrence of events, while the former primarily associates with finding model parameters for observed data. Both serve important usage in the current industry and are significant for business growth, such as applying likelihood vs. probability in risk assessment.

Understanding the distinction between likelihood and probability is paramount in data analysis and decision-making. Probability quantifies the likelihood of an event based on available information, while likelihood assesses the plausibility of different parameters given the observed data. Both concepts are indispensable in statistical modeling and inference.

Moreover, recognizing the significance of likelihood and probability is crucial in decision-making. By acquiring foundational knowledge in data science and AI, non-technical professionals can gain the ability to make informed decisions. Our No-code AI program democratizes access to data analytics, empowering learners to embrace data-driven decision-making confidently. It is an excellent choice for professionals seeking to integrate data science and AI into their daily work lives.

Frequently Asked Questions