"Striking the Balance: Bias and Variance in Machine Learning and the Golf Analogy"

Topics that we are going to discuss Bias variance tradeoff is. 

what is 

Bias 

variance

overfitting

underfitting

bias variance tradeoff

bias variance decomposition

The Hidden Truth

To understand the bias-variance tradeoff, let's start with the example you provided. You have a dataset consisting of the CGPA, IQ, and LPA (salary) of 1000 college students. Your goal is to build a machine learning model that can predict the LPA of a given student based on their CGPA and IQ data.

In order to establish a mathematical relationship between the input variables (CGPA and IQ) and the target variable (LPA), you can use a linear regression model, which assumes a linear relationship between the variables. 

The model equation would look like this: LPA = b0 + b1 * CGPA + b2 * IQ.

However, since you only have a sample of data and not the entire population, you need to make predictions or estimations about the population data. In other words, you want to find a model that closely approximates the true mathematical relationship between the variables, denoted as y = f(x) + irreducible error.

The irreducible error is the naturally occurring error that cannot be eliminated or reduced. It captures factors beyond the scope of your model, such as measurement errors or unobserved variables. We cannot do anything about this error, so our goal is to minimize the reducible error.

To approximate the true relationship, we aim to find a formula or mathematical relationship that closely resembles f(x). Let's call this approximation y' = f'(x). When we make predictions using our model, we obtain y_hat, which is not exactly equal to the true y, 

introducing an error term: 

reducible error = f(x) - f'(x), which can also be expressed as (y - y_hat).

The bias-variance tradeoff is about finding the right balance between bias and variance in our model to minimize this reducible error. Bias refers to the error introduced by approximating a real-world problem with a simplified model. A high bias model tends to underfit the data, meaning it oversimplifies the relationships and may not capture important patterns. On the other hand, variance refers to the model's sensitivity to fluctuations in the training data. A high variance model tends to overfit the data, meaning it captures noise or random fluctuations instead of the underlying patterns.

In the context of the bias-variance tradeoff, we want to reduce the reducible error by managing both bias and variance. By increasing model complexity, we can reduce bias and better capture intricate relationships in the data. However, this may lead to higher variance and an increased tendency to overfit. On the contrary, by reducing model complexity, we can decrease variance but risk introducing higher bias and underfitting the data.

Reducible error=bias²+var

The goal is to find the sweet spot where the model has low bias and low variance, striking a balance between underfitting and overfitting. This can be achieved through techniques such as regularization, cross-validation, or ensemble methods that combine multiple models. By understanding and managing the bias-variance tradeoff, we can improve the predictive performance of our machine learning models.

The "trade-off" in bias-variance trade-off refers to the fact that minimizing bias will usually increase variance and vice versa.

Some questions

1. How would you define bias and variance mathematically?
2.  How is bias and variance related to overfitting and underfitting mathematically? 
3. Why is there a tradeoff between bias and variance mathematically?

Expected value is nothing but population mean not exactly rougly speaking 

Discreate random variable 

Let's start with the expected value formula for a discrete random variable.

For a discrete random variable X that can take on a finite or countably infinite number of values, the expected value (also known as the mean or average) is calculated as:

E(X) = Σ (x * P(X = x))

In this formula:

Σ represents the summation symbol, indicating that we need to sum over all possible values of X.

x represents each possible value that X can take.

P(X = x) represents the probability of X taking the value x.

To compute the expected value, we multiply each possible value of X by its corresponding probability, and then sum up these products.

Now, let's move on to the expected value formula for a

continuous random variable.

For a continuous random variable X, which can take on any value within a continuous range, the expected value is computed using an integral rather than a summation.

The expected value (mean) of a continuous random variable X with probability density function (PDF) f(x) is given by:

E(X) = ∫ (x * f(x)) dx

In this formula:

∫ represents the integral symbol, indicating that we need to integrate over the entire range of X.

x represents the variable of integration, which takes on values within the range of X.

f(x) represents the probability density function of X.

To calculate the expected value, we multiply each value of x by its corresponding probability density (f(x)), and then integrate over the entire range of X.

It's important to note that the expected value represents the average or central tendency of the random variable, providing a measure of its long-term average behavior.

The variance of a population, denoted as Var(X), measures the spread or variability of a random variable X around its expected value. It quantifies how much the values of X deviate from their average.

The formula to compute the variance of a population is:

Var(X) = E[X^2] - (E[X])^2

Let's break down the meaning and derivation of this formula:

Expected value (E[X]):

The expected value, E[X], is the average value or the mean of the random variable X. It represents the central tendency or the typical value that X takes.

Expected value of squared values (E[X^2]):

E[X^2] is the expected value of X squared. It measures the average of the squared values of X.

Derivation of the variance formula:

To derive the formula for variance, we start with the definition of variance as the average squared deviation from the mean.

Var(X) = E[(X - E[X])^2]

Expanding the square term, we get:

Var(X) = E[X^2 - 2X * E[X] + (E[X])^2]

Now, let's distribute the expectation operator over the expanded terms:

Var(X) = E[X^2] - 2 * E[X * E[X]] + E[(E[X])^2]

Since E[X * E[X]] can be written as E[X] * E[X], we have:

Var(X) = E[X^2] - 2 * E[X]^2 + E[(E[X])^2]

Simplifying further, we notice that E[(E[X])^2] is equal to (E[X])^2:

Var(X) = E[X^2] - 2 * E[X]^2 + (E[X])^2

Combining like terms, we obtain the final formula for variance:

Var(X) = E[X^2] - (E[X])^2

This formula tells us that the variance of a random variable X is equal to the expected value of X squared minus the square of

the expected value of X. It represents the average squared deviation of X from its mean.

BIAS VARIANCE DECOMPOSTION

Bias-variance decomposition is a way of analysing a learning algorithm's expected generalization error with respect to a particular problem by expressing it as the sum of three very different quantities: bias, variance, and irreducible error. 

1. Bias: This is the error from erroneous assumptions in the learning algorithm. High bias can cause an algorithm to miss the relevant relations between features and target outputs (underfitting). 
2. Variance: This is the error from sensitivity to small fluctuations in the training set. High variance can cause an algorithm to model the random noise in the training data, rather than the intended outputs (overfitting). 
3. Irreducible Error: This is the noise term. This part of the error is due to the inherent noise in the problem itself, and can't be reduced by any model.

loss=reducible error irreducible error

loss=bias²+variance+irerecuiable error 

INTITUTION 

Certainly! Let's use the example of a golfer taking shots to illustrate the intuition behind bias and variance.

Imagine a golfer who wants to hit a golf ball towards a specific spot on the course. The golfer represents our machine learning model, and each shot the golfer takes corresponds to a different model trained on the same dataset but with some variability.

In this context, bias refers to the golfer's average deviation from the desired spot. If the golfer consistently hits shots that are far away from the target spot, it indicates a high bias. On the other hand, if the golfer's shots tend to cluster tightly around the target spot, the bias is low.

Variance, in this analogy, represents the spread or variability in the golfer's shots. If the golfer's shots are scattered all over the course, it indicates high variance, suggesting that the golfer's performance is highly inconsistent. Conversely, if the shots are consistently close to each other, the variance is low.

To reduce bias, the golfer needs to apply more energy and make adjustments in their swing technique. Similarly, in machine learning, to reduce bias, we need to use more complex models or algorithms that can capture intricate patterns in the data. For example, using polynomial regression or decision trees with more depth can help reduce bias.

However, increasing the complexity of the models to reduce bias may lead to increased variance. In our golf analogy, if the golfer puts too much energy into their shots, they may hit the ball with different angles and directions, causing the shots to spread out more. Similarly, in machine learning, complex models can be sensitive to small variations in the training data, resulting in higher variance.

To reduce variance, regularization techniques can be applied. Regularization adds a penalty term to the model's objective function, discouraging overly complex models. This helps to reduce the variability and makes the model more robust to variations in the training data.

In summary, the bias-variance tradeoff suggests that as we try to reduce bias, we may increase variance, and vice versa. Finding the right balance depends on the specific problem and the available data. The golfer analogy helps illustrate the idea that reducing bias may require more energy or complexity, while reducing variance may require regularization or techniques to control the variability in the model's predictions.

Loss can be breakdown into bias²+variance+irreduciable error.