quality:corruption, linguistic:voice_change
Making a prediction

An archaeologist is working at a site thought to be from the medieval period, between the 11th century to the 16th century. However, it is uncertain exactly when in this period the site was inhabited. Fragments of pottery are found, some of which are glazed and some of which are decorated. It is expected that if the site were inhabited during the early medieval period, then 1% of the pottery would be glazed and 50% of its area decorated, whereas if it had been inhabited in the late medieval period then 81% would be glazed and 5% of its area decorated. How confident can the archaeologist be in the date of inhabitation as fragments are unearthed?

The degree of belief in the continuous variable C {\displaystyle C} (century) is to be calculated, with the discrete set of events { G D, G D ¯, G ¯ D, G ¯ D ¯ } {\displaystyle \{GD,G{\bar {D}},{\bar {G}}D,{\bar {G}}{\bar {D}}\}} as evidence. Assuming linear variation of glaze and decoration with time, and that these variables are independent,

P (E = G D ∣ C = c) = (0.01 + 0.81 − 0.01 16 − 11 (c − 11)) (0.5 − 0.5 − 0.05 16 − 11 (c − 11)) {\displaystyle P(E=GD\mid C=c)=(0.01+{\frac {0.81-0.01}{16-11}}(c-11))(0.5-{\frac {0.5-0.05}{16-11}}(c-11))} P (E = G D ¯ ∣ C = c) = (0.01 + 0.81 − 0.01 16 − 11 (c − 11)) (0.5 + 0.5 − 0.05 16 − 11 (c − 11)) {\displaystyle P(E=G{\bar {D}}\mid C=c)=(0.01+{\frac {0.81-0.01}{16-11}}(c-11))(0.5+{\frac {0.5-0.05}{16-11}}(c-11))} P (E = G ¯ D ∣ C = c) = ((1 − 0.01) − 0.81 − 0.01 16 − 11 (c − 11)) (0.5 − 0.5 − 0.05 16 − 11 (c − 11)) {\displaystyle P(E={\bar {G}}D\mid C=c)=((1-0.01)-{\frac {0.81-0.01}{16-11}}(c-11))(0.5-{\frac {0.5-0.05}{16-11}}(c-11))} P (E = G ¯ D ¯ ∣ C = c) = ((1 − 0.01) − 0.81 − 0.01 16 − 11 (c − 11)) (0.5 + 0.5 − 0.05 16 − 11 (c − 11)) {\displaystyle P(E={\bar {G}}{\bar {D}}\mid C=c)=((1-0.01)-{\frac {0.81-0.01}{16-11}}(c-11))(0.5+{\frac {0.5-0.05}{16-11}}(c-11))}

Assume a uniform prior of f C (c) = 0.2 {\textstyle f_{C}(c)=0.2}, and that trials are independent and identically distributed. When a new fragment of type e {\displaystyle e} is discovered, Bayes' theorem is applied to update the degree of belief for each c {\displaystyle c}: f C (c ∣ E = e) = P (E = e ∣ C = c) P (E = e) f C (c) = P (E = e ∣ C = c) ∫ 11 16 P (E = e ∣ C = c) f C (c) d c f C (c) {\displaystyle f_{C}(c\mid E=e)={\frac {P(E=e\mid C=c)}{P(E=e)}}f_{C}(c)={\frac {P(E=e\mid C=c)}{\int _{11}^{16}{P(E=e\mid C=c)f_{C}(c)dc}}}f_{C}(c)}

A computer simulation of the changing belief as 50 fragments are unearthed is shown on the graph. In the simulation, the site was inhabited around 1420, or c = 15.2 {\displaystyle c=15.2}. By calculating the area under the relevant portion of the graph for 50 trials, the archaeologist can say that there is practically no chance the site was inhabited in the 11th and 12th centuries, about 1% chance that it was inhabited during the 13th century, 63% chance during the 14th century and 36% during the 15th century. The Bernstein-von Mises theorem asserts here the asymptotic convergence to the "true" distribution because the probability space corresponding to the discrete set of events { G D, G D ¯, G ¯ D, G ¯ D ¯ } {\displaystyle \{GD,G{\bar {D}},{\bar {G}}D,{\bar {G}}{\bar {D}}\}} is finite (see above section on asymptotic behaviour of the posterior).