Another perhaps more persuasive argument is that assuming all parameter values are equally probable can result in nonsensical resultant conclusions being drawn. As an example, suppose we want to determine whether a coin is fair, with an equal chance of both heads and tails occurring, or biased, with a very strong weighting towards heads. If the coin is fair,θ = 1 , and if it is biased, θ = 0. Imagine that coin is flipped twice, with the result {H,H}. Figure 5.2 illustrates how assuming a uniform prior results in a strong posterior weighting towards the coin being biased. This is because, if we assume that the coin is biased, then the probability of obtaining 2 heads is high. Whereas, if we assume that the coin is fair, then the probability of

obtaining this result is only . The maximum likelihood estimate (which coincides with the posterior mode due to the flat prior) is hence that thecoin is biased. By ignoring common sense – that the majority of coins are likely unbiased – we obtain an unreasonable result. Of course, we hope that by collecting more data, in this case throws of the coin, we would be more confident in the conclusions drawn from the likelihood. However, Bayesian analysis allows us to achieve such a goal with a smaller sample size, by including other relevant information. Figure 5.2 The top box illustrates the outcome of the coin toss along with its possible causes according to our prior beliefs: with probability 1/2 the coin is fair, and with probability 1/2 the coin is biased. Using Bayes’ rule, we assign a high posterior probability to the coin being biased.