Abstract
Bayesian inference has a long standing history in the world of statistics and this chapter aims to serve as an introduction to anyone who has not been formally introduced to the topic before. First, Bayesian inference is introduced using a simple and analytical example. Then, computational methods are introduced. Examples are provided with common HCI problems such as comparing two group rates based on a binary variable, numeric variable, as well as building a regression model.
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- 1.
Priors with higher variance can be considered less informative in this setting.
- 2.
A more formal version of the likelihood would be \(p(\{y_1,...,y_n\}|\theta ) = \prod _{i} \theta ^{y_i} (1-\theta )^{(1-y_i)}\), where the set \(D = \{y_1,...,y_n\}\) represents the outcome for the sequence of attempts to turn on the device (Kruschke 2013).
- 3.
In the Beta/Binomial approach, the prior is defined using the Beta distribution’s probability density function (PDF). The simplified form of Beta’s PDF (for this type of problem), is \(p(\theta |\alpha ,\beta ) \propto \theta ^{\alpha -1} (1-\theta )^{\beta -1}\). Assuming that the friend told the child that he/she has seen these devices turn on ten times (\(\alpha = 10\)) and fail to turn on two times (\(\beta = 2\)), our prior would be: \(p(\theta |\alpha = 10, \beta = 2) \propto \theta ^{10-1} (1-\theta )^{2-1}\). The likelihood function is based on the Bernoulli distribution with 1 successes and 0 failures expressed as \(p(D|\theta ) \propto \theta ^{1} (1-\theta )^{0}\). Using Bayes’ Rule we can combine the likelihood and prior to produce the posterior distribution: \(p(\theta |D) \propto p(D|\theta ) p(\theta ) = \theta ^{10-1} (1-\theta )^{2-1} \theta ^{1} (1-\theta )^{0} = \theta ^{10} (1-\theta )^{1}\). The posterior density is a beta density that we can easily interpret if we calculate its \(\alpha \) and \(\beta \) parameters: \(\alpha =10+1\) and \(\beta =1+1\). As such the mean for \(\theta \) is \(M = \alpha / (\alpha +\beta ) = 11/(11+2) = 0.846\) or the child’s beliefs that the device will turn on is focused at 84.6 %. The standard deviation is \(SD = \sqrt{\frac{\alpha \beta }{(\alpha +\beta )^2(\alpha +\beta +1)}} \approx 0.0093\). The probability interval with a 95 % probability will be \(0.846 \pm 1.96 \times 0.0093\) which places the child’s belief in the device turning on between 82.7 % and 86.4 %.
- 4.
An alternative approach to solving the problem would be to use Bayesian Probit Regression Jackman 2009.
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Tsikerdekis, M. (2016). Bayesian Inference. In: Robertson, J., Kaptein, M. (eds) Modern Statistical Methods for HCI. Human–Computer Interaction Series. Springer, Cham. https://doi.org/10.1007/978-3-319-26633-6_8
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