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Volume 2 - LENR-CANR

Volume 2 - LENR-CANR

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Figure 8. Uniform prior P(Hp)<br />

Here P(E11 | Hp) is p, because that’s what Hp says: the probability of getting a head is p. And<br />

P(Hp) is 1 by assumption.<br />

The continuous version of the sum rule (§2.1.2) gives<br />

as in Fig. 9.<br />

1<br />

E11 PE11<br />

| H p PH <br />

P p<br />

<br />

0<br />

1<br />

<br />

p dp 1/<br />

2<br />

dp<br />

0<br />

| E11<br />

(5)<br />

H p<br />

P p 2<br />

Figure 9. One head in one trial observed<br />

Now the probability of heads on the next trial is:<br />

1<br />

P(“one more head” | E11) = P<br />

(“one more head” | E11Hp) P(Hp | E11) dp<br />

0<br />

The first factor in the integrand is p by definition of Hp, and equation (5) gives the second. So<br />

2<br />

P(“one more head” | E11) = 2<br />

p dp 2/<br />

3<br />

1<br />

0<br />

We can continue making trials and updating our probability distribution for Hp. Possible<br />

results are shown in Figs. 10–12.<br />

713

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