混合分布

Mixture distributions are statistical models that represent a population composed of multiple underlying subpopulations, each of which is represented by its own probability distribution. The overall model is a weighted sum (or mixture) of these component distributions. Each component can be thought of as representing a different 'source' within the overall population, and the weights reflect the proportion of each component in the mixture.

Key points about mixture distributions:

1. **Components:** A mixture distribution is made up of several component distributions. These could be normal distributions, Poisson distributions, or any other type, potentially even different types mixed together.

2. **Weights:** Each component distribution has an associated weight (or mixing proportion) that indicates the fraction of the overall population that comes from that component. These weights must sum to 1.

3. **Probability Density Function (PDF):** The PDF of a mixture distribution is the sum of the PDFs of the component distributions, each weighted by its respective weight. For a mixture of \( k \) components, the PDF is:
\[ f(x) = \sum_{i=1}^{k} w_i f_i(x) \]
where \( w_i \) is the weight for the \( i \)-th component distribution with PDF \( f_i(x) \).

4. **Flexibility:** Mixture distributions can model a wide variety of data that cannot be adequately represented by a single distribution. For example, they can capture multimodality (data with multiple peaks).

5. **Applications:** Mixture models are used in various fields such as finance (to model returns), biology (to model heterogeneity in populations), and machine learning (as in Gaussian Mixture Models used in clustering).

6. **Estimation:** The parameters of mixture distributions (the parameters of each component and the weights) are often estimated using the Expectation-Maximization (EM) algorithm, which iteratively finds estimates that maximize the likelihood of the observed data.

An example of a mixture distribution is a Gaussian mixture model, where all components are normal distributions with different means and variances. This model can fit complex data structures by adjusting these parameters and weights to capture the underlying patterns in the data.