What is Bayesian Thinking?

As students, we often ponder how our results will be after the final term examinations. So, we start speculating based on our previous internal marks performance, the number of all-nighters we have pulled, and our prior performance in similar courses. This approach of updating our beliefs about our potential performance aligns very closely with a powerful statistical framework known as “Bayesian Thinking”. This technique adopts the logic of Bayesian theorem which we know in machine learning as the Bayes formula. You might’ve never quite realized it, but most of our introspection regarding the future is heavily dependent on Bayes’ conditional probability. In this article, we will dive deeper into how we can correlate Bayesian thinking with our daily life to formalize and improve our estimations of future outcomes.

Core of Bayesian Thinking

Bayesian thinking, as the name suggests, is based on the Bayes Theorem, which predominantly follows these 3 fundamental concepts – prior, likelihood, and posterior. Let’s understand them based on the example of gauging our final exam performance.

1. Prior: The initial belief we have upon an uncertain (e.g., the probability you’ll get an A on the final exam) before seeing new data.
2. Likelihood: The probability of understanding new data given a particular hypothesis (e.g., how likely we are going to score well in the final exam if we study x hours per day).
3. Posterior: The updated belief we have when a new situation occurs, which is calculated using Bayes’ Theorem.

Here, for 2 events A and B:

P(A) is the prior probability of hypothesis B.
P(B|A) is the likelihood of data B given A.
P(B) is the marginal probability of data B.
P(A|B) is the posterior probability of A after observing B.

So, in our exam-based scenario:

• Hypothesis (H): The thought that “I will achieve grades like 85-90% in the final exam.”
• Data (D): Information available before the final, like remaining study hours, internal exam scores, difficulty of past topics, number of modules, etc.
• Prior: Our initial belief about scoring 85-90% based on past performance (e.g., previous final exams, overall CGPA, etc.).
• Likelihood: What are the chances of achieving the observed internal score if you are truly an 85-90% performer.
• Posterior: Our updated belief about the chance of scoring 85-90% after considering internal performance and remaining study days.

Why Use Bayesian Thinking?

Now that you understand what Bayesian thinking is, let me tell you how it helps in decision-making and why we need to use it.

1. Modeling Uncertainty: In simple words, this means our gut feeling about how we have performed in the exam. Bayesian thinking forces us to quantify our uncertainties, such as assuming getting a score between 83-85. This can lead us to better decision-making.
2. Fusing Multiple Evidences: We can systematically collect diverse information like past grades, past year FAQs, etc. The evidence here can be considered as our independent features.
3. Dynamic Updation: As we gather more information like the effectiveness of group study or referring to a topper’s notes, etc., we will update our posterior, which later becomes our new prior for the next evidence.
4. Better Planning and Resource Allocation: If our posterior probability of an A grade is still low despite all the extra studying, we might shift our focus to the next optimal grade – B, by putting more effort into our weak modules and optimizing our plan.

Understanding the Scenario Better

Let’s dive deeper into understanding how our exam scenario plays out by integrating all the following Bayes’ conditional probabilities. In this case, our calculation would be as follows:

1. Setting up the Prior

Imagine you’re a third-year engineering student with a historical average score of 75% in your major subjects. Based on your overall academic record, you may believe there is:

• A 25% chance of scoring >=90% (A Grade)
• A 50% chance of scoring 80-90% (B Grade)
• A 25% chance of scoring 70-80% (C Grade)

The percentages we have made above make up the prior distributions across our performance bands. We are to follow the Bayes Formula fundamental concepts to map out our values here.

Here these values can be considered as our Bayesian conditional probabilities or distributions.

2. Gathering New Evidence

Two weeks before the final, you receive your internal exam result which is 80%. How should this affect your belief about the final? First, we gotta estimate the likelihood:

• Say you truly are an A‑level performer (≥ 90%), historically you score more than 80% on internals 80% of the time.
• Say you’re a B‑level performer (80–90%), you score more than 80% on internals 40% of the time.
• Say you’re a C‑level performer (70–80%), you rarely score that high, maybe about 10% of the time.

3. Computing the Evidence Probability

To normalize and compute P(D), the overall probability of scoring 80% on the internal would be as follows:

P(D)=(0.80×0.25)+(0.40×0.50)+(0.10×0.25)
P(D) = 0.20+0.20+0.025=0.425

4. Calculating the Posterior

Here we will be applying Bayes’ theorem for each band:

P(A∣D)=(0.80×0.25) / 0.425 ≈ 0.47
P(B∣D)=(0.40×0.50) / 0.425 ≈ 0.47
P(C∣D)=(0.10×0.25) / 0.425 ≈ 0.06

As you can see, the results show:

• 47% chance of being an A‑level performer,
• 47% chance of B‑level,
• 6% chance of C‑level.

5. Incorporating Study Effort

The following week, you log and track your daily study hours. Let’s say the historical data suggests that you study ≥ 5 hours/day in the last 2 weeks. Now,

• An A‑level student typically follows this 70% of the time.
• A B‑level student, 30% of the time.
• A C‑level student, 5% of the time.

Suppose you averaged 6 hours/day. This becomes another piece of data ‘S’, for which we will need to compute the updated likelihoods:

We will be utilizing the Bayesian formula here in a loop for each updation of our belief as newer evidence occurs. Normalize with P(S):

P(S)=(0.70×0.47)+(0.30×0.47)+(0.05×0.06) ≈ 0.329+0.141+0.003=0.473

Upon further updation:

P(A∣D,S)=0.70×0.47 / 0.473​ ≈ 0.70
P(B∣D,S)=0.30×0.47 / 0.473 ​≈ 0.30
P(C∣D,S)=0.05×0.06​ / 0.473 ≈ 0.01

Your belief in getting an A‑grade rises to 70% after accounting for your diligent study.

6. Considering Remaining Days

Now, let’s go with the assumption that there are 7 days left before the final exam, each being an opportunity to revise or reinforce learning. Suppose, mastering the remaining topics translates into an extra 5 percentage marks on the final with:

• 70% likelihood for an A‑level student who studies intensely,
• 30% for a B‑level student,
• 5% for a C‑level student.

Normalize and update one more time. The final posterior would be like:

P(A ∣ all) ≈ 0.84
P(B ∣ all) ≈ 0.16
P(C ∣ all) ≈ <0.01

The final posterior shows a 75% chance of getting an A, 24% for B, and <1% for C. Based on this, our overall percentage is very likely to increase.

If you happen to come from an ML background, I’m pretty sure you might find this article pretty familiar. Yes, we are following the very same mechanism that is utilized in Naive Bayes, which is the Bayes Formula. For those who don’t know Naive Bayes, here are 2 articles that can help you learn about it:

• An Introduction to the Powerful Bayes’ Theorem for Data Science Professionals
• Naive Bayes Classifier Explained With Practical Problems

Making Decisions Based on Bayesian Thinking

With a posterior distribution over our performance bands, we can now make sound and optimized decisions. Here’s how:

• Targeted Revision: If your chance of getting an A remains marginal (say 55%), focus on high-yield topics that escalate you from B to A, rather than wasting more time on well-mastered material.
• Risk Management: If your chance of getting a B is high but an A is slim, ensure you secure partial credit on challenging questions to lock in the B. This will help ensure you get more time on optimizing your time and resources for other subjects that have a higher yield of getting an A.
• Resource Allocation: Decide whether investing extra hours in group study or topper’s notes makes the most sense, by estimating how much such interventions shift the posterior.

Practical Tips for Applying Bayesian Thinking

Bayesian thinking doesn’t quite require complex maths. We just need a clear, structured approach to updating our beliefs when we get our new pieces of evidence. Whether you’re making decisions in your personal life, work, research, or learning, viewing your progress as a dynamic system of beliefs shaped by data, can lead to more informed and smarter decision-making.

Here are some practical ways to apply Bayesian reasoning in everyday scenarios:

• Quantify Your Priors: Start by reflecting on what you already know and assign rough probabilities (we take estimates since we can’t be exact) to possible outcomes.
• Gather Reliable Likelihood Estimates: Look for historical patterns or correlations relevant to your situation. If personal data isn’t available, seek insights from similar experiences, trusted peers, or domain experts. This information can be gathered from others’ experiences too.
• Track Evidence Methodically: Keep a record of meaningful observations, feedback, results from small experiments, etc., so that each new piece of data can be factored into updating your beliefs.
• Use Simple Tools: A basic spreadsheet can be maintained to keep track of how your prior beliefs evolve with each piece of new evidence. Labeling each step can make the updating process more transparent and manageable.
• Update Frequently, but Thoughtfully: Don’t overreact to noise or minor fluctuations. Instead, choose logical checkpoints (like weekly reviews, milestones, or key decisions) for formal updates to your beliefs.
• Interpret Posteriors in Context: A 60% probability of success may be encouraging, but not definitive. Use these updated probabilities to guide your actions, while continuing to refine your strategies and seek new evidence.

Applications of Bayesian Thinking

While our example centers on exam performance, Bayesian reasoning applies universally. Some common applications include:

• Medical Diagnosis: Doctors update disease probabilities as test results arrive.
• Machine Learning: Bayesian models treat parameters as distributions, enabling principled uncertainty estimation.
• Business Forecasting: Firms adjust sales projections as new market data flows in.
• Everyday Life: Even deciding whether to carry an umbrella or not, given a weather forecast and current sky conditions, is a form of Bayesian thinking.

By consciously framing problems in terms of priors, likelihoods, and posteriors, we gain more clarity and adaptability in our decision-making. We can quantify how much new information can alter our minds, avoiding overreaction to noise or underreaction to crucial evidence.

You can read more about Hidden Markov Models here.

Conclusion

Bayesian thinking turns any uncertainty into a clear, transparent, and optimized decision-making process. Defining your initial assumptions, assessing how new information or features would alter them, and continuously updating this data can help you cultivate both clarity and confidence in your decisions. Whether you’re evaluating project outcomes, medical diagnoses, market trends, or everyday choices, mastering this approach provides a powerful framework for decision‑making under uncertainty. Next time you face an unknown, lean on your priors, weigh your evidence, and let Bayes’ theorem guide you through to reach a more informed judgment.