DDE-1/Statistics.md

+_C. Ates_
+
+[[_TOC_]]
+
+In the lecture, we will use probability-based learning in various types of problems. here you can find the summaries of key terminologies and the concepts, with additional references. 
+
+## Statistical Inference
+
+### Basics
+
+Lets imagine that you are collecting data via your observations. In order to talk about probabilities, we inherently assume that there is a space, a sample space which engulfs every single possible outcomes. For coin flipping, the practical possibilities are either head, or tail. So sample space has 2 elements. Any subset of the sample space is called event. 
+
+With these basic definitions, we can construct the concept of probability, that is the probability of an event to occur: $`p(E) `$.  If the whole sample space probability is scaled to 1 ($`p(S)=1`$), then any event has the natural probability limits ranging $`0<=p(E)<=1`$. With this notation, we can describe the probabilities of the different sub-sample spaces. For instance, for a regular dice, the probability to observe either 2 or 3 is $`p(E={2,3})=1/3`$. Here, we naturally perform the following operation:
+
+```math
+p(E={2,3})= p({2})+ p({3}) - p({2,3}) = p({2})+ p({3})
+```
+
+as the events are mutually exclusive. 
+
+In this lecture, we will be using conditional probabilities a lot in probability-based learning. We will be interested in what is the probability of observing j, given that i occured. This is different that the probability of observing both i and j:
+
+```math
+p(E_j|E_i)= p(E_j)p(E_i)/ p(E_i)
+```
+
+where | denotes the conditional statement. Herein, the dependecy of the events is important to us. We want to know whether a set of observations  $`E_1`$ is correlated with a second set of observations $`E_2`$. As a matter of fact, we are intersted in finding such patterns in the data. 
+
+Two events are said to be independent if:
+
+```math
+p(E_1E_2)= p(E_1)p(E_2)
+```
+
+or we can say:
+
+  ```math
+p(E_1E_2)= p(E_1)p(E_2)
+```
+
+p(E_2|E_1)= p(E_2); p(E_1|E_2)= p(E_1)
+```
+
+### Bayesian statistics
+
+...
+
+### What is likelihood?
+
+...