Statistics 101: Bayes Theorem and its real life application

 Introduction

The ‘Bayes' Theorem’ is a conditional probability that assesses uncertainty in the context of a particular situation. The Bayes' rule is applied in a variety of situations, such as when diagnosing a rare disease. By using the Bayes' rule, we may calculate the likelihood that the condition for which the test was designed truly exists.

For example, you can be friends with A given that you are friends with B can be deduced by using Bayes Theorem. This takes into consideration the probability that you are friends with B given that you are already friends with A, multiplied by the probability that you are simply friends with A. To rule out flaws in this calculation, we further divide the whole calculation with the probability that you are simply friends with B, to complete our application of the theorem.

In addition to specific situations, Bayes' theorem may be used in daily situations like dating and friendships. You may improve your understanding of the selection effects and alter your actions or behaviors to create more of the connections you believe are most beneficial by taking an honest look at your present friendships or romantic relationships. You may evaluate what you gain and what you lose from doing particular actions by using Bayes' theorem.

The Bayes theorem is a crucial component of sophisticated machine learning models and inference statistics. Bayesian inference plays a crucial role in research since it provides a rational method for updating the potential of hypotheses in light of new information. It provides an explanation of the likelihood of an event based on prior knowledge of potential relevant conditions. The Bayes theorem offers a formula for determining the level of uncertainty. It may be used in everyday circumstances where we're trying to decide something based on fresh knowledge.

Moreover, the Bayes Theorem is a mathematical paradigm that uses statistics and probability to calculate the likelihood of a situation based on how it relates to other possible outcomes. That is the most crucial guideline for making wise decisions. The Bayes theorem draws conclusions based on the information at hand and takes "conditional probabilities" into account. Prior probabilities may result in posterior probability distributions that are skewed.

Real-time application of Bayes’ theorem

Following are some of the real-time and most relevant applications of Bayes’ theorem:

1. Model Animal Behaviour - Animals must behave based on shaky information and with little cognitive capacity in order to live. It is generally known that our sensory and sensorimotor processing uses probabilistic estimates to get around these restrictions. Probabilistic estimating does, in fact, explain how animals learn, forage, and make mate decisions. The theoretical concerns of how animals combine previous knowledge and observations are fundamentally based on the Bayes theorem. Yet, given the right posterior probability, we are more interested in whether animals make the best judgements rather than what conclusions can be reached about unknown parameters.

This might be, for instance, the distribution of food patch characteristics or mate potential. The animal is thus thought to be capable of developing a "posterior opinion" based on sampling information, such as the standard of a certain food patch or the typical attributes of mates over the course of a year. Their priors may originate from either one of two sources, or from both: either their own individual experience obtained from sampling the environment, or past generations' adaptation to the environment. As a result, we might anticipate frequently witnessing "Bayesian-like" decision-making in nature. Foraging in patches, choosing a partner during an annual breeding season, and growing in the face of predator risk are some biological examples.

1. Weather Forecasting - The pillars of observation, information acquisition, and prediction form the basis of the scientific method. The quality of our existing knowledge and the accuracy of our observations affect how accurate our projections will be. Weather predictions are a typical example; the more we know about how the weather works, the better we can predict whether it will rain tomorrow using current observations and seasonal records, and any disparity between prediction and observation may be utilized to improve the weather model.

Consider the scenario where we want to determine the probability of rain when Alexa forecasts rain. Instead of the probability that Alexa would properly forecast rain on days when it has already rained. The conditional probability of rain if Alexa anticipated rain may be expressed mathematically as P(Rain|RainPrediction) and interpreted as such:

• P(Rain), or the likelihood that it will rain
• P(RainPrediction), which is the likelihood that Alexa will predict that it will rain
• P(RainPrediction|Rain), which is the likelihood that a rain forecast will be made on days when it does in fact rain.
• P(Rain|RainPrediction), which is the likelihood that it will rain on days for which we have a rain forecast.

1. Cardiovascular nursing - When faced with uncertainty and unpredictability, healthcare choices for complicated situations are typically left to the discretion of nurses. This decision-making typically relies on educated predictions supported by the data at hand. Also, nurses are being required to order and interpret diagnostic tests more often, needing a comprehension of the significance of estimating the likelihood of particular outcomes. Probabilities are used to depict chance or a numerical estimate of the uncertainty connected to an occurrence or events. Using a probabilistic technique, one may determine the level of uncertainty surrounding a given result, such as a diagnosis or test result. Thus, the application of Bayesian statistics in clinical nursing judgment and decision-making is beneficial.

To ascertain if the information found may be employed as reliable information in the form of a prior distribution, the axiomatic basis of Bayesian premises enable direct comparisons of nursing diagnoses in distinct realities. Regardless of sample size, the use of Bayesian approaches in cardiovascular nursing research can be beneficial for analysing unusual occurrences and the likelihood of diagnosis in particular groups. Moreover, the Bayesian paradigm deals with the idea of uncertainty, making it applicable to methods that seek to establish elements like accuracy, sensitivity, and specificity.

1. Predicting environmental damage - Whether or not the detected pollutant concentrations exceed the water quality standard numerical limits determines whether or not a water body is suitable for its intended use (such as drinking water, recreation, or agricultural usage). As the main pollutant can't always be tested directly, scientists often study markers that act as potential substitutes for it. The connection between an indicator concentration and the concentration of the pollutant it is meant to represent varies significantly depending on the kind of pollutant.

In this context, by first applying a prior probability distribution to the population parameters and (which may account for a priori beliefs about their potential values), then creating a likelihood function for and based on empirical evidence (in this case, water quality samples), and finally deriving a joint posterior probability distribution for both, we can explicitly acknowledge many uncertainties in a Bayesian framework.

1. Machine learning - In the area of machine learning, the Bayes Theorem is also widely applied. We can compute P(B|A) in terms of P(A|B), P(B), and P(A) thanks to the Bayes theorem (A). This rule is helpful when trying to figure out the fourth term when there is a strong likelihood that P(A|B), P(B), and P. (A). The Naive Bayes classifier, which is used in classification algorithms to separate data based on accuracy, speed, and classifications, is the simplest implementation of the Bayes theorem. Let's have a look at a machine learning application of the Bayes theorem.

Consider that vector A has the property I. It denotes that A = A1, A2, A3, A4,...... Ai

‘n’ classes are also represented by the letters C1, C2, C3, C4,..........Cn.

We have been given these two criteria, and our machine-learning-based classifier must forecast A, with the best class being the default. As a result, we may phrase it as follows using Bayes' theorem:

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In this case, P(Ci/A) = [P(A/Ci) * P(Ci)] / (A)

P is the condition-unaffected entity (A).

Conclusion

As was already said, the use of Bayes' Theorem in the field of medicine may be seen when there is ambiguity in the diagnosis, choice of therapy, and prognosis prediction. It may be used in the fields of finance and business, machine learning, and spam filtering to determine if an email is spam or not, as well as to forecast weather, anticipate animal behavior, and predict environmental harm. It also applies to areas like dating. By looking at your present friendships or partners, you may better grasp the consequences of selection and alter your actions or behaviors to create more of the connections you value the most.

The Bayes theorem is also used in areas including psychology, bioinformatics, legal applications, and finding lost goods. The Bayes theorem offers a straightforward method for thinking and computing. It lets us make predictions about the event under examination or research using knowledge from the past and observations. In Bayesian inference, all unknowns in a system are modeled by using Bayes' theorem to update probability distributions as more information comes to light. While Thomas Bayes created it in the 18th century, we can see that it still has numerous advantages for humans today.

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