A Natural Nuclear Reactor — Question 10

Question 10

Independent isotope signatures strengthen a hypothesis quantitatively

For a hypothetical evidence exercise, let H mean ancient fission and A an alternative geological process. Priors are $P(H)=.010$ and $P(A)=.990$. A uranium anomaly has likelihood ratio 10 in favor of H; a stable-product anomaly has ratio 20. Assume conditional independence under both hypotheses.

1. Update prior odds with both observations and calculate the posterior probability of H.

2. Calculate the posterior using only the uranium anomaly. Why can correlated observations not automatically be multiplied?

Question 10 - Solution

Independent isotope signatures strengthen a hypothesis quantitatively

(a) Method and calculation

$$O_{\mathrm{prior}}=P(H)/P(A)=1/99,\quad\boxed{O_{\mathrm{post}}=(1/99)(10)(20)=200/99}.$$$$\boxed{P(H\mid E_1,E_2)=\frac{200}{200+99}=.668896}.$$

(b) Interpretation and further calculation

$$\boxed{P(H\mid E_1)=\frac{10}{10+99}=.0917431}.$$$$\frac{P(E_1,E_2\mid H)}{P(E_1,E_2\mid A)}=\frac{P(E_1\mid H)P(E_2\mid E_1,H)}{P(E_1\mid A)P(E_2\mid E_1,A)}.$$

Physical check

The likelihoods are invented teaching data, not a historical confidence estimate for Oklo. Multiple signatures require attention to dependence and geological context.