4.4 Bayesian hypothesis testing

A more radical way of including previous knowledge in statistical inference is Bayesian inference. Bayesian inference regards the sample that we draw as a means to update the knowledge that we already have or think we have on the population. Our previous knowledge is our starting point and we are not going to just discard our previous knowledge if a new sample points in a different direction, as we do when we reject a null hypothesis.

Think of Bayesian inference as a process similar to predicting the weather. If I try to predict tomorrow’s weather, I am using all my weather experience to make a prediction. If my prediction turns out to be more or less correct, I don’t change the way I predict the weather. But if my prediction is patently wrong, I try to reconsider the way I predict the weather, for example, paying attention to new indicators of weather change.

Bayesian inference uses a concept of probability that is fundamentally different from the type of inference presented in previous chapters, which is usually called frequentist inference. Bayesian inference does not assume that there is a true population value. Instead, it regards the population value as a random variable, that is, as something with a probability.

Again, think of predicting the weather. I am not saying to myself: “Let us hypothesize that tomorrow will be a rainy day. If this is correct, what is the probability that the weather today looks like it does?” Instead, I think of the probability that it will rain tomorrow. Bayesian probabilities are much more in line with our everyday concept of probability than the dice-based probabilities of frequentist inference.

Remember that we are not allowed to interpret the 95% confidence interval as a probability (Chapter 3)? We should never conclude that the parameter is between the upper and lower limits of our confidence interval with 95 per cent probability. This is because a parameter does not have a probability in frequentist inference. The credible interval (sometimes called posterior interval) is the Bayesian equivalent of the confidence interval. In Bayesian inference, a parameter has a probability, so we are allowed to say that the parameter lies within the credible interval with 95% probability. This interpretation is much more in line with our intuitive notion of probabilities.

Bayesian inference is intuitively appealing but it has not yet spread widely in the social and behavioral sciences. Therefore, I merely mention this strand of statistical inference and I refrain from giving details. Its popularity, however, is increasing, so you may come in contact with Bayesian inference sooner or later.