Demo: Widgetized coin tossing

6.3. Demo: Widgetized coin tossing#

In this section we consider an interactive version of the problem of estimating the probability that a particular coin will come up heads on any given toss, \(p_H\), based on data as to how many heads (and tails) it produces in \(N\) tosses. The tosses are assumed to be independent. (This example is gratefully adapted from the similar example in Sivia’s book.)

The data \(D\) will then be the number of heads \(H\) obtained in \(N\) trials. The probability of obtaining a particular number of heads will be a function of \(p_H\). This is the likelihood piece of Bayes’ theorem. Note that the outcome is discrete (either heads or tails), and the number of heads obtained in \(N\) trials is an integer, but \(p_H\) can be any real number \(0 \leq p_H \leq 1\), and all our output pdfs are continuous functions of \(p_H\) in the interval \(0 < p_H < 1\).

Meanwhile we can represent different prior knowledge and/or beliefs about \(p_H\) in the prior, i.e., \({\rm p}(p_H|I)\). \(I\) could be information regarding the character of the coin flipper, it could be based on a previous experiment (we managed to get hold of the coin and flip it a few times before hand!), or it could be an “ignorance prior”, the formulation of which we will come back to later in the book.

Bayesian updating#

One of the key points of this exercise, is that with each flip of the coin we acquire more information on the value of \(p_H\). The logical thing to do is to update the state of our belief, our pdf for \(\mathrm{p}(p_H|\mbox{no. tosses, no. heads},I)\) each time the number of coin tosses is incremented by 1. The pdf will tend to get narrower as we acquire more data, i.e., our state of knowledge of \(p_H\) becomes more definite.

Note that in what follows we exploit the fungibility of mathematical symbols to let \(I\) stand for different things at different stages of the coin tossing experiment. If we are going to “update” after every coin toss then \(D\) is just the result of the \(N\)th coin toss and \(I\) is what we know about the coin after \(N-1\) coin tosses.

User-interface for coin-flipping#

Take a look at the information under the Help tab to find out about what the controls do, what the priors are, etc.

Widget user interface features:

  • tabs to control parameters or look at documentation

  • set the true \(p_H\) by the slider

  • press “Next” to flip “jump” # of times

  • plot shows updating from three different initial prior pdfs

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