Guide to probability Density Estimation & Maximum Likelihood Estimation

Density Estimation is an essential part of both machine learning and statistics. It means getting the probability density function (PDF) of a group. It is necessary for many things, like finding outliers, putting things into groups, making models, and finding problems. Based on deep learning, this study looks at all the ways to measure old and new density.

Traditional Density Estimation Methods

Histograms

Whether you need to know in a hurry whether your data collection is complete, a histogram is the way to go. They take the data range and chunk it up into categories called " bins " to determine the frequency distribution of events. The height of each bin shows how many people are expected to live in that group.

Kernel Density Estimation (KDE)

Kernel Density Estimation (KDE) is a method that doesn't use factors to figure out how dense a data set is. When you put a kernel function at each data point and add them all up, you get a smooth density measure. How well KDE works depends a lot on which kernel function you choose and how much smoothing you set with the bandwidth option.

Gaussian Mixture Models (GMM)

GMM is based on the idea that the data come from a mix of Gaussian models. Fitting the data with a weighted sum of Gaussian components lets it work out the density. The number of features and their values are figured out using a stepwise expectation-maximization method.

Parametric Density Estimation Methods

Parametric Models

In parametric density forecasts, the data is supposed to fit a certain framework. Maximum Likelihood Estimation (MLE) is a way to figure out the numbers of the factors that make it most likely that the data will be seen. In machine learning, MLE is often used to fit parametric models and figure out what the parameters are. It requires making the probability function, making it bigger (often by using the log-probability), and finding out the numbers. MLE is used for things like linear regression, logistic regression, and Gaussian mixture models. It lets you guess, take samples, and figure out how the information is spread.

Mathematical Formulation of MLE

The likelihood function L(θ) reflects the chance of witnessing the data under the model given a statistical model with parameters and a collection of independent and identically distributed (i.i.d.) observations x1, x2,..., xn. Assuming that the observations are selected from the model's probability distribution, the likelihood function may be defined as the joint probability of the observations −

$\mathrm{L(\theta) \: = \: P(x_{1},x_{2},\dotso , x_{n}| \theta)}$

MLE aims to find the parameter values that maximize the likelihood function L(θ). This can be formulated as −

$\mathrm{\theta \: = \: \arg\max_{t}\:L(\theta)}$

In practice, it is often more convenient to work with the log-likelihood function, given by −

$\mathrm{\ell(\theta) \: = \: \log \: L(\theta)}$

Because the logarithm is a monotonically growing function, maximizing the log-likelihood is identical to maximizing the likelihood function.

Estimating Parameters using MLE

MLE lets us figure out what the parameters are by setting the derivatives of the log-likelihood function to zero concerning the parameters. This makes a set of formulae that can be used to find the best numbers for the factors.

For example, Think about fitting a Gaussian distribution to some data. The probability function is equal to the sum of the individual Gaussian probabilities −

$\mathrm{L(\mu ,\: \sigma^{2}) \: = \: \Pi_{i}P(x_{i}\: | \: \mu, \: \sigma^{2})\: = \: \Pi_{i} \: 1 /(\surd (2\pi \: \sigma^{2}))\: ^{*} \: \exp(−(x_{i} \: − \: \mu)^{2}\:/\:(2\sigma^{2}))}$

When we use the logarithm, we get −

$\mathrm{\ell(\mu ,\: \sigma^{2}) \: = \: \Sigma_{i}[\log(1 /(\surd (2\pi \: \sigma^{2})))\: − \: (x_{i} \: − \: \mu)^{2}\: / \: (2\sigma^{2})]}$

To estimate the parameters $\mathrm{\mu}$ and $\mathrm{\sigma^{2}}$ , we differentiate $\mathrm{\ell(\mu ,\: \sigma^{2})}$ concerning $\mathrm{\mu}$ and $\mathrm{\sigma^{2}}$ and set the derivatives to zero. Solving these equations gives the maximum likelihood estimates for $\mathrm{\mu}$ and $\mathrm{\sigma^{2}}$.

Properties of MLE

MLE possesses several desirable properties −

• Consistency − As the sample size increases, the MLE gets closer to the actual parameter values under certain situations.

• Efficiency − The MLE is asymptotically efficient, achieving the smallest possible asymptotic variance among consistent estimators.

• Asymptotic Normality − The MLE has a normal distribution with the actual parameter values in the middle. This feature makes it possible to create confidence ranges and test hypotheses.

Application in Machine Learning

In machine learning, MLE is often used to predict the parameters of different models, such as linear regression, logistic regression, hidden Markov models, Gaussian mixture models, and many others. It gives a way to fit models to data based on principles and easy to do on a computer.

Conclusion

Determining how dense something is is one of the most important parts of machine learning. You can make guesses close to the truth using traditional tools like histograms, kernel density estimates, and Gaussian mixture models. New methods like mixed-density networks, variational autoencoders, and changing flows are more flexible and have led to great results based on a deep knowledge. Maximum Likelihood Estimation (MLE) is a popular technique in both fields. As a result, we may use the information we have to determine the model's parameters.. This shape is good, average, and steady.