Linear Regression

In statistics, linear regression is an approach for modeling the relationship between a scalar dependent variable $y$ and one or more explanatory variable denoted $X$.

In other words, we try to fit the data using a line.

$y = kx + b$

which can best represent the trainning data. We can write the function as following:

$h(x) = \theta_0 + \theta_1x$

In real world the planning is difficult. Because of the environment is:

1. Stochastic
2. Multi-agent
3. Partial observability

Instead of planning in the world states, we planning it in the belief state.

IID

We assume the data is IID which means identically distributed and independently drawn from same distribution.

Mixture models

Recall types of clustering methods

1. hard clustering: clusters do not overlap
   • element either belongs to cluster or it does not
2. soft clustering: clusters may overlap
   • stength of association between clusters and instances

Mixture models (soft clustering)

1. probabilistically-sounded way of doing soft clustering
2. each cluster: a generative model (Gaussian or multinomial)
3. parameters(e.g. mean/covariance are unknown)

Probabilistic Inference

Hidden variable is the variable we don’t need to know, don’t need to output, but we need to compute during the process.

Eumeration

Eumeration goes through all the possibilities, adds them up and comes with an answer.