Probability
To understand reinforcement learning, the first
concept you need to grasp is probability. According to its dictionary
definition (Wikipedia), probability is a method of expressing knowledge or
belief about whether an event will occur or has occurred. Probability also
refers to the rate at which specific outcomes occur given the same cause. When
you think of probability, the concept of a dice game might come to mind.
A die has six faces numbered from 1 to 6. The probability of rolling a 1 is
1/6, something we understand intuitively. However, if you actually buy a die
from a store and roll it six times, will a 1 definitely come up once? Not
necessarily. A probability of 1/6 means that if you roll the die an infinite
number of times, on average, a 1 will come up once every six rolls. Therefore,
the concept of probability is intertwined with randomness. When we say
something is probabilistic, we can think of it as random. However, like in the
dice game, probability can be calculated based on the number of possible
outcomes.
Conditional Probability
Conditional probability refers to a special case of
probability that describes the likelihood of an event occurring under a
specific condition. For example, the probability of event B occurring given
that event A has occurred is denoted as P(B|A), and this is called the
conditional probability of event B given event A. Let's assume we have a
classroom with 5 male students and 5 female students.
Conditional Probability
(https://pixabay.com/)
Now, let's say that 2 male students have laptops, and
3 female students have laptops. The probability of selecting a student with a
laptop from the entire class would be 5/10, which is 1/2. However, the
probability that a male student has a laptop is 2/5. In this case, event A is
selecting a male student, and event B is selecting a student with a laptop. In
other words, we are referring to the probability of selecting a student with a
laptop under the condition that the student is male. So, what is the
probability that a female student has a laptop? It is easy to see that the
probability is 3/5.
Stochastic Process
A stochastic process is a combination of the
concept of stochastic (probability) and process (a sequence of
steps). As mentioned earlier, stochastic refers to something that may appear
random in the short term but follows certain rules over a longer period. A process
is something that is related to the passage of time. Processes like growth,
development, and evolution are determined by the flow of time. Therefore, a
stochastic process refers to a state that moves randomly (probabilistically)
over time.
Stochastic Process
There are various ways to mathematically express a
stochastic process, but a common representation is {Xt}. Here, X
represents a random variable, and t represents time. The curly braces {}
denote a set, meaning that a stochastic process can be represented as a set of
random variables occurring over time.
The concept of a stochastic process was created to
solve specific problems. To solve a scientific concept, the first step is to
represent the phenomenon mathematically. If a phenomenon can be mathematically
represented, it can be programmed, making problem-solving easier. Therefore, a
stochastic process can be defined as a mathematical representation of a state
or environment that changes randomly over time.
A prominent example of a stochastic process is Brownian
motion. Brownian motion is a phenomenon discovered in 1827 by the Scottish
botanist Robert Brown, who theoretically explained the irregular movement of
pollen particles on the surface of water. Previously, it was believed that only
living organisms could move by themselves, but Robert Brown demonstrated that
the same irregular motion occurred even when using inanimate materials like
dust or glass.
Examples of Brownian
Motion (https://en.wikipedia.org/wiki/Brownian_motion)
If pollen starts at a specific point and moves
randomly at regular time intervals, after n movements, the distance from
the starting point can be measured. If n is large enough, the
probability of where the pollen will be located can be calculated.
This phenomenon, discovered by Robert Brown, was
further elaborated by Einstein. Einstein showed that the cause of Brownian
motion is the collision of liquid molecules and formalized this phenomenon in
mathematical terms.
The concept of Brownian motion is widely used in
fields like statistical mechanics and economics. In economics, Brownian motion
is often used to explain the rules that govern market movements.
