Distinguishing between two types of coins.

<p>On the top there are two possible coins with slightly different distributions for yielding a head (<i>H</i>) or a tail (<i>T</i>). (We depicted two possible outcomes but our model can account for more.) Given a sequence of observations (corresponding to the random outcomes of coin tosses), the goal of the observer is to guess the coin type being used (either 0 or 1). The wear induced by tossing the coins may, with time, change the probability that they land on either heads or tails in a way that depends on the coin type as well as on the previous toss outcomes (observations). In particular, notice that without a change in the probability of heads in our example we would not obtain a posterior probability 0.51 starting with a prior of 0.45 after six coin flips involving two heads. <i>P</i>_<i>j</i>(<i>H</i>^(<i>t</i>) ∣ observations) for <i>j</i> ∈ {0, 1} denotes the probability of Coin of type <i>j</i> to yield <i>H</i> given the particular sequence of <i>t</i> observations. Here, this sequence is <i>H</i>, <i>T</i>, <i>T</i>, <i>H</i>, <i>T</i>, <i>T</i>. At the beginning of the next round, <i>i.e.</i>, the 7th round, |<i>ε</i>^{(<i>t</i> = 6)}|₁ measures how “far” the the <i>H</i> vs <i>T</i> distribution generated by the worn down coin 1 is from the same distribution as generated by the worn down coin 2. More precisely, for the case above, |<i>ε</i>⁽⁶⁾|₁ = |<i>P</i>₀(<i>H</i> ∣ observations) − <i>P</i>₁(<i>H</i> ∣ observations)| + |<i>P</i>₀(<i>T</i> ∣ observations) − <i>P</i>₁(<i>T</i> ∣ observations)| = |0.51 − 0.43| + |0.49 − 0.57| = 0.16. The parameter <i>ε</i> bounds all possible |<i>ε</i>^(<i>t</i>)|₁ from above.