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Market regime after big moves

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The market is in a high-volatility regime with probability $0.2$ and in a low-volatility regime with probability $0.8$ . On each day, given the regime, a big move occurs independently with probability $0.7$ in the high regime and $0.1$ in the low regime. You observe big moves on $2$ consecutive days. What is the probability that the regime is high? Give a fraction in lowest terms.

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Hint

Multiply the prior of each regime by the likelihood of the $2$ observed days, then normalize.

Solution

For the high regime the joint weight is $0.2 \times 0. 7^{2} = 0.2 \times 0.49 = 0.098$ .
For the low regime the joint weight is $0.8 \times 0. 1^{2} = 0.8 \times 0.01 = 0.008$ .
The total weight is $0.098 + 0.008 = 0.106$ .
The posterior probability of the high regime is $\frac{0.098}{0.106}$ , which is $\frac{49}{53}$ .

Answer: $\frac{49}{53}$

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