Definition 3: Non-frequentist subjective approach

The frequentist approach has a number of disadvantages. Firstly, it can not be used to provide probability estimates for events that occur once only or rarely (e.g. climate change). Secondly, the frequentist estimates are based ENTIRELY on the sample and so can not take into account any prior belief (common sense) about the probability. For example, an unbiased coin could easily produce 2 heads only when tossed 10 times and this would lead to a frequentist probability estimate of 0.2 for heads. However, our belief in the rarity of biased coins would lead us to suspect this estimate as being too low. In other words, the frequentist estimate does not really reflect our true beliefs in this case.

In such cases a more subjective approach to probability must be adopted that takes into account ALL the available information. The subjective probability of an event A can be defined as the price you would pay for a fair bet on the event divided by the amount you would win if the event happens. Fair means that neither you or the bookmaker would be expected to make any net profit. To make a fair bet all the prior information must be taken into account - e.g. the biasedness of coins, the previous form of a horse in a horse race, etc. This can be done most conveniently by making use of Bayes' theorem (covered later in section 3.5 of this chapter). The Bayesian approach takes a more relativist view of probability and instead uses data to update prior probability esimates to give improved posterior probability estimates.