A Brief Look at Bayesian Statistics

 

There are two main strands of statistics, Classical statistics and Bayesian statistics. These aren’t necessarily conflicting ideologies, though many statisticians throughout history would beg to differ, but are simply two different ways to tackle a problem. Hopefully this post will give you some brief insight into the uses and differences of the two approaches.

Classical statistics is the first type of statistics that people come across and is to do with what we expect to happen in a repeatable experiment. This might be the idea that if we flip a coin an infinite number of times the proportion of heads we obtain is a half. Hence we get the well known probability of a head as a 1/2. This is why classical statistics is often known as frequentist and covers ideas such as confidence intervals and p-values.

Bayesian statistics evolved out of Bayes’ Theorem which I talked about in a previous post.

P(A|B) = \cfrac{P(B|A)P(A)}{P(B)}

  • P(A), P(B) are known as prior probabilities, because we know them before we learn any more information.
  • P(A|C), P(B|C) are known as posterior probabilities, because they are found after we have learnt some additional information.

You can think of this Bayesian statistics as an evolution of Bayes’ Theorem. Instead of dealing with point probabilities we now deal with probability distributions. As a result we now have prior and posterior distributions to consider.

f(\theta|x) = \cfrac{f(x|\theta)f(\theta)}{f(x)}

As the term f(x) is just a normalising constant we can drop it to get the commonly seen Bayes’ Rule.

f(\theta|x) \propto f(x|\theta)f(\theta)

Here f(\theta|x) is the posterior distribution, f(x|\theta) is the likelihood which accounts for the statistical model and f(\theta) is the prior which represents the expert beliefs before seeing the data. The key point is that the Bayesian approach can quantify theories and hypotheses, something that can be desirable.

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Why Lawyers Need Statistics

I previously talked about Bayes’ theorem and its often misunderstood applications. Normally these mistakes aren’t particular costly or harmful in the world of statistics, but if they are used to make decisions that impact on the real world then getting things wrong can be extremely costly.

One place where statistics can be called upon to influence important matters is the court. Throughout the last 50 years there has been an increase in the use of statistics in court matters and it is important that everybody involved understands them and their use. If any or all of the prosecution, defence or jury misinterpret the information given to them then the chances of a miscarriage of justice will greatly increase.

\text{Prob of matching a description} \neq \text{Prob of matching a description and being guilty}

The classical mistake made in the past by many prosecuting teams is that of the ‘prosecutors fallacy‘. This is when the prosecution or defence have presented the jury with some statistic such as a probability that has been calculated incorrectly, yet manage to convince the jury to accept its truth.

Scales1

Scales of justice; source

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What is Bayes’ Theorem?

One of the most famous theorems in statistics and probability is that of Bayes’ Theorem, which first appeared around 250 years ago. It allows us to calculate reverse probabilities and use new evidence to update our beliefs. For example the probability of a hypothesis given a set of evidence can be found from the probability of that evidence given a hypothesis.

To understand Bayes’ Theorem it is important to have a basic understanding of conditional probability. This is the probability of something happening given that some event has already happened. Some examples of conditional probabilities are given below,

  • Given that Watford scored a goal, what was the probability that Odion Ighalo scored?
  • Given that it rained yesterday, what is the probability that it will remain tomorrow?
  • Given a sports centre has a swimming pool, what is the probability it also has a gym?

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