# The Frequentist vs. Bayesian in Sports Betting

###### John Alesia

##### Frequentist vs. Bayesian in Sports Betting

In sports betting, as in other areas of statistical analysis, there are two main approaches to estimating the probability of an event occurring: the frequentist approach and the Bayesian approach. These approaches differ in how they handle uncertainty and how they update their beliefs about the probability of different events occurring.

The frequentist approach is based on the idea of repeated sampling and assumes that the probability of an event occurring is fixed, but unknown. In this approach, the probability of an event occurring is estimated based on the frequency with which the event occurs in a large number of samples or trials. For example, suppose we want to estimate the probability of Team A winning a hockey game. In that case, we could gather data on the outcomes of a large number of previous games and calculate the proportion of games that Team A won. This proportion would be our estimate of the probability of Team A winning the game.

The Bayesian approach, on the other hand, is based on the idea of subjective probability and allows for the incorporation of prior beliefs and additional information as it becomes available. In this approach, the probability of an event occurring is viewed as a degree of belief that is updated as new data is collected. For example, suppose we want to estimate the probability of Team A winning a football game. In that case, we could start with a prior belief about the probability of this event occurring based on factors such as the teams' past performance, current form, and injuries. We could then use Bayes' theorem to update this belief as we collect more data about the game, such as the teams' recent marks and injuries.

Overall, the main difference between the frequentist and Bayesian approaches in sports betting is that the frequentist approach estimates the probability of an event occurring based on the frequency of the event in a large number of samples. In contrast, the Bayesian approach allows for incorporating prior beliefs and additional information as it becomes available. Both approaches have their strengths and limitations and can be used to make informed predictions about the outcomes of games and to improve the chances of making successful bets.

##### Example of Bayesian approach in sports betting

Suppose we want to make a bet on the outcome of a basketball game between Team A and Team B. We can start by defining a hypothesis about each team's probability of winning the game. This hypothesis, also known as a prior belief, can be based on various factors such as the teams' past performance, current form, and injuries.

For example, let's say our initial prior belief about the probability of Team A winning the game is 0.4, and our initial prior belief about the probability of Team B winning the game is 0.6. These prior beliefs could be based on our analysis of the team's past performance, current form, and injuries, as well as any additional information available to us.

Once we have defined our prior beliefs about the probability of each team winning the game, we can use Bayes' theorem to update these beliefs as we collect more data. Bayes' theorem states that the posterior probability of a hypothesis (in this case, the probability of a team winning the game) is equal to the prior probability of the hypothesis multiplied by the likelihood of the data given the hypothesis, divided by the probability of the data. This can be written mathematically as:

P(H|D) = (P(D|H) * P(H)) / P(D)

where P(H|D) is the posterior probability of the hypothesis (the probability of a team winning the game) given the data (information about the game such as the teams' past performance, current form, and injuries), P(D|H) is the likelihood of the data given the hypothesis (the probability of the data occurring given the probability of a team winning the game), P(H) is the prior probability of the hypothesis (our initial belief about the probability of a team winning the game), and P(D) is the probability of the data (the probability of the data occurring).

As the game progresses, we can use Bayes' theorem to update our beliefs about the probability of each team winning the game based on new information that becomes available. For example, if Team A takes an early lead and is ahead by 10 points at halftime, we might update our belief about the probability of Team A winning the game. If we plug these values into the equation above, we might get:

P(H|D) = (P(D|H) * P(H)) / P(D) = (0.9 * 0.4) / (0.9 * 0.4 + 0.1 * 0.6) = 0.36 / 0.46 = 0.78

This means that our updated belief about the probability of Team A winning the game is 0.78, which is higher than our initial belief of 0.4. Based on this updated belief, we might decide to make a bet on Team A to win the game with a live bet if the odds justify it.

Of course, this is just a simple example, and in practice, we would likely have more data and a more complex model to estimate the probability of each team winning the game. However, the basic idea is the same: we start with a prior belief about the probability of an event occurring (in this case, a team winning a game), and use Bayes' theorem to update this belief as we collect more data. By doing so, we can make more informed predictions about the outcomes of games and improve our chances of making successful bets.