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192CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCYThe updated probabilities p 0t =P(δ 0 |∆f t ,...,∆f 1 ) are called the posteriorprobabilities. An equivalent way to obtain the posterior probabilitiesisp 0,1 =p 0,2 =.p 0t =p 0 P(∆f 1 |δ 0 )p 0 P(∆f 1 |δ 0 )+p 1 P(∆f 1 |δ 1 ) ,p 0,1 P(∆f 2 |δ 0 )p 0,1 P(∆f 2 |δ 0 )+p 1,1 P(∆f 2 |δ 1 ) ,p 0,t−1 P(∆f t |δ 0 )p 0,t−1 P(∆f t |δ 0 )+p 1,t−1 P(∆f t |δ 1 ) .How long is the learning period? To start things off, you need to specifyan initial prior probability, p 0 =P(δ = δ 0 ). 11 Let δ 0 =0, δ 1 =1,andlet v have a discrete probability distribution with the probabilities,P(v = −5) =66 3 P(v = −1) =11 2 P(v =3)=663P(v = −4) =66 3 P(v =0)=11 2 P(v =4)=663P(v = −3) =66 3 P(v = 1) =11 2 P(v =5)=663P(v = −2) =11 1 P(v =2)=111We generate the distribution of posterior probabilities, learningtimes, and forecast error autocorrelations by simulating the economy2000 times. Figure 6.3 shows the median of the posterior probabilitydistribution when the initial prior is 0.95. The distribution of learningtimes and autocorrelations is not sensitive to the initial prior. Thelearning time distribution is quite skewed with the 5, 50, and 95 percentilesof the distribution of learning times being 1, 14, and 66 periodsrespectively. Judging from the median of the distribution, Bayesianupdaters quickly learn about the true economy. Since the forecast errorsare serially correlated only during the learning period, we calculatethe autocorrelation of the forecast errors only during the learning period.The median autocorrelations at lags 1 through 4 of the forecast11 Lewis’s approach is to assume that learning is complete by some date T>t 0 inthe future at which time p 0,T = 0. Having pinned down the endpoint, she can workbackwards to Þnd the implied value of p 0 that is consistent with learning havingbeen completed by T .