Statistical inference is the process of drawing conclusions about a population using a sample. In practical analytics work, this often means choosing a model and estimating its parameters so that the model explains the observed data as well as possible. One of the most widely used techniques for this is Maximum Likelihood Estimation (MLE). MLE is the method of selecting parameter values that maximise the probability of observing the data you actually collected. If you are building strong foundations through a data analyst course in Bangalore, MLE is a core idea that connects probability theory to real modelling decisions.
What MLE Is Trying to Solve
Most models include unknown parameters. For example, a normal distribution has a mean (μ) and standard deviation (σ). A logistic regression model has coefficients that control how features influence the probability of a class. The key question is: which parameter values best fit the data?
MLE answers this using a simple principle: assume a model form, then choose parameters that make the observed data most “likely” under that model. Here, “likelihood” has a specific meaning. It is not a probability of parameters. Instead, it treats the observed data as fixed and the parameters as variables to be chosen.
This matters because it provides a consistent rule for estimation across many model types, from basic distributions to complex machine learning models.
Likelihood vs Probability: The Intuition
Probability is usually written as P(data∣θ)P(\text{data} \mid \theta)P(data∣θ), where θ\thetaθ represents parameters. If you plug in a parameter value, you can compute how probable the data would be under that setting.
Likelihood flips your perspective. You keep the data fixed and view the same expression as a function of θ\thetaθ:
L(θ)=P(data∣θ)L(\theta) = P(\text{data} \mid \theta)L(θ)=P(data∣θ)The “best” parameters are the ones that maximise L(θ)L(\theta)L(θ). In words: choose the model settings under which the data you observed would be most plausible.
In real analytics tasks, this idea shows up everywhere. When you fit regression models, estimate conversion rates, or tune probabilistic classifiers, you are often performing MLE under the hood. This is why a data analyst course in Bangalore typically includes MLE early, before moving to broader inference tools.
How MLE Works Step by Step
Although the math can vary by model, the workflow is consistent:
- Choose a model and its parameter(s).
- Example: assume data comes from a normal distribution with parameters μ and σ.
- Write the likelihood of the observed sample.
- If observations are independent, the likelihood is the product of each point’s probability (or density) under the model:
- L(θ)=∏i=1nf(xi∣θ)L(\theta) = \prod_{i=1}^{n} f(x_i \mid \theta)L(θ)=i=1∏nf(xi∣θ)
- Use the log-likelihood for convenience.
- Products become sums, which are easier to optimise:
- ℓ(θ)=logL(θ)=∑i=1nlogf(xi∣θ)\ell(\theta) = \log L(\theta) = \sum_{i=1}^{n} \log f(x_i \mid \theta)ℓ(θ)=logL(θ)=i=1∑nlogf(xi∣θ)
- Maximise the log-likelihood.
- You either:
- take derivatives and solve (closed-form solutions exist for many classical models), or
- use numerical optimisation (common for logistic regression and more complex models).
This approach is powerful because it scales. It can be applied to small textbook examples and to large real-world datasets with millions of rows.
A Simple Example: Estimating a Coin’s Bias
Suppose you flip a coin 100 times and get 62 heads. You want to estimate the probability of heads, ppp. Under a Bernoulli model, the likelihood of observing the sequence is proportional to:
L(p)=p62(1−p)38L(p) = p^{62}(1-p)^{38}L(p)=p62(1−p)38To find the maximum likelihood estimate, you choose ppp that maximises L(p)L(p)L(p). If you take logs and differentiate, the result is:
p^=62100=0.62\hat{p} = \frac{62}{100} = 0.62p^=10062=0.62So the MLE is simply the sample proportion. This example is useful because it shows how MLE often aligns with intuitive estimators.
In business analytics, similar logic appears when estimating click-through rates, conversion rates, defect rates, and churn probabilities, common topics in a data analyst course in Bangalore, where learners work with binomial outcomes.
Practical Considerations and Common Pitfalls
MLE is widely used, but it is not the automatic “truth.” A few practical points matter:
- Model assumptions drive results.
- If you assume normality but the data is heavily skewed, the MLE parameters may be misleading. MLE can only be as good as the model class you choose.
- Outliers can affect estimates.
- Some likelihoods are sensitive to extreme values. For example, estimating σ in a normal model can be pulled upward by outliers.
- Small samples can produce unstable estimates.
- In limited data settings, MLE can have high variance. This is one reason analysts sometimes use regularisation or Bayesian approaches.
- Multiple maxima can exist.
- Some models have likelihood surfaces with local maxima, so optimisation methods must be chosen carefully.
Understanding these issues helps you interpret model outputs instead of treating them as black-box numbers.
Why MLE Matters in Modern Analytics
MLE is more than a statistical technique. It is a unifying idea behind many tools used in analytics and machine learning. Logistic regression, Naïve Bayes, many time-series models, and even parts of deep learning training can be framed as likelihood maximisation (often via minimising negative log-likelihood).
Once you understand MLE, you also gain a clearer view of related concepts like confidence intervals, hypothesis tests, and model comparison, because many of them build on likelihood behaviour.
Conclusion
Maximum Likelihood Estimation is a foundational method in statistical inference: it selects parameter values that make the observed data most likely under a chosen model. The process, define a likelihood, takes logs, optimise, and appears across probability models and practical machine learning workflows. By learning how MLE works, when it succeeds, and where it can fail, you become better at building and evaluating models rather than simply running them. This is exactly why MLE remains a key topic in applied learning paths, such as a data analyst course in Bangalore, where understanding data-driven decision-making matters as much as producing a final metric.
