advanced-algorithmic-trading

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This is in contrast to another form of statistical inference, known as Classical or Frequentist
statistics, which assumes that probabilities are the frequency of particular random events
occuring in a long run of repeated trials.
For example, as we roll a fair unweighted six-sided die repeatedly, we would see that each
number on the die tends to come up 1/6th of the time.
Frequentist statistics assumes that probabilities are the long-run frequency of random
events in repeated trials.
When carrying out statistical inference, that is, inferring statistical information from probabilistic
systems, the two approaches–Frequentist and Bayesian–have very different philosophies.
Frequentist statistics tries to eliminate uncertainty by providing estimates. Bayesian statistics
tries to preserve and refine uncertainty by adjusting individual beliefs in light of new evidence.
2.1.1 Frequentist vs Bayesian Examples
In order to make clear the distinction between these differing statistical philosophies, we will
consider two examples of probabilistic systems:
• Coin flips - What is the probability of an unfair coin coming up heads?
• Election of a particular candidate for UK Prime Minister - What is the probability
of seeing an individual candidate winning, who has not stood before?
Table 2.1.1 describes the alternative philosophies of the frequentist and Bayesian approaches.
In the Bayesian interpretation probability is a summary of an individual’s opinion. A key
point is that various rational, intelligent individuals can have different opinions and thus form
alternative prior beliefs. They have varying levels of access to data and ways of interpreting it.
As time progresses information will diffuse as new data comes to light. Hence their potentially
differing prior beliefs will lead to posterior beliefs that converge towards each other, under the
rational updating procedure of Bayesian inference.
In the Bayesian framework an individual would apply a probability of 0 when they believe
there is no chance of an event occuring, while they would apply a probability of 1 when they are
absolutely certain of an event occuring. Assigning a probability between 0 and 1 allows weighted
confidence in other potential outcomes.
In order to carry out Bayesian inference, we need to utilise a famous theorem in probability
known as Bayes’ rule and interpret it in the correct fashion. In the next section Bayes’ rule
is derived using the definition of conditional probability. However, it isn’t essential to follow the
derivation in order to use Bayesian methods, so feel free to skip the following section if you
wish to jump straight into learning how to use Bayes’ rule.
Note that due to the presence of cognitive biases many individuals mistakenly equate highly
improbable events with events that have no chance of happening. This manifests in common
vernacular when individuals state that certain tasks are "impossible", when in fact they may be
merely very difficult. In quantitative finance this is extremely dangerous thinking, as it ignores
the ever-present issue of tail-risk. Consider the failures of Barings Bank in 1995, Long-Term
Capital Management in 1998 or Lehman Brothers in 2008. In Bayesian probability this usually
translates as applying very low probability in priors to "impossible" chances, rather than zero.