高校数学の質問スレ（医者・東大卒専用） Part438

高校数学の質問スレ（医者・東大卒専用） Part438 (979ﾚｽ)
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581: １３２人目の素数さん [sage] 2025/01/17(金) 18:02:26.71 ID:BcOp0/e9

# incubation period for COVID-19
incu=\(x){dlnorm(x,m=1.64,s=0.58)
# lm=runif(1,1.6,1.8)
# s=runif(1,0.4,0.6)
}
incu=Vectorize(incu)
# window period for rapid antigen test
wind=\(x) dweibull(x, shape=2.5,scale=4.5)
wind=Vectorize(wind)
curve(wind,0,14,col=2,xlab='days',ylab='',axes=F) ; axis(1)
curve(incu,add=TRUE)
legend('top',bty='n',legend=c('incubation period','window period'),lty=1:1,col=1:2)

cdf_incu=\(x) integrate(incu,0,x)$value
cdf_incu=Vectorize(cdf_incu)
cdf_wind=\(x) integrate(wind,0,x)$value
cdf_wind=Vectorize(cdf_wind)
curve(cdf_wind,0,14,col=2)
curve(cdf_incu,add=TRUE)
legend('center',bty='n',legend=c('incubation period','window period'),lty=1:1,col=1:2)

pdfZ=\(z) integrate(\(x) incu(x)*wind(x-z),0,Inf)$value
pdfZ=Vectorize(pdfZ)
curve(pdfZ,-10,10,xlab='incubation-window (days)',axes=F,ylab='') ; axis(1)
integrate(pdfZ,-Inf,0)$value
fnr=\(x) integrate(pdfZ,-Inf,-x)$value # false negative rate after x days since onset of symtom
fnr=Vectorize(fnr)
curve(fnr,0,7, xlab='days after onset',ylab='false negative rate')

##
k=1e5
inc=rlnorm(k,m=1.64,s=0.58)
win=rweibull(k,shape=2.5,scale=4.5)
d=inc-win
hist(d)
mean(inc)
mean(win)
mean(inc<win) # negative test
re=NULL
for(i in 1:7) re=c(re,mean((inc+i)<win)) ; re
plot(re,pch=16)
which(re<0.05)[1]

http://rio2016.5ch.net/test/read.cgi/math/1723152147/581

856: １３２人目の素数さん [sage] 2025/05/30(金) 20:13:47.71 ID:zoAuXcvc

p_post_null <- function(p0, prior, alpha, beta, success, failure){
# Calculate the total number of trials from successes and failures.
total = success + failure

# Calculate the likelihood of the data under the null hypothesis (H0).
# This assumes a binomial distribution where the success probability is p0.
m0 = dbinom(success, total, p0)

# Calculate the marginal likelihood of the data under the alternative hypothesis (H1).
# Under H1, the success probability is assumed to follow a Beta distribution
# with parameters alpha and beta. This function (from the 'extraDistr' package)
# computes the marginal likelihood by integrating over all possible success probabilities.
m1 = extraDistr::dbbinom(success, total, alpha, beta)

# Calculate the Bayes Factor (BF01).
# This is the ratio of the likelihood under the null hypothesis (m0)
# to the marginal likelihood under the alternative hypothesis (m1).
BF01 = m0 / m1

# Calculate the posterior probability of the null hypothesis.
# This updates the prior belief (prior) based on the evidence from the data (BF01).
p_post = prior * BF01 / (prior * BF01 + 1 - prior)

# Return both the Bayes Factor and the posterior probability as a named vector.
c(BF01 = BF01, p_post = p_post)
}

# Optimize the 'alpha' parameter to maximize the posterior probability of the null hypothesis.
# We're trying to find the 'alpha' value (within the range 0 to 1e6) that makes the
# null hypothesis most plausible, given the data and the relationship beta = 5*alpha - 4.
# p_post_null(...)[2] specifically extracts the 'p_post' value from the function's output.
optimize(function(alpha) p_post_null(p0 = 1/6,
prior = 1/2,
alpha,
5 * alpha - 4, # Beta parameter is a function of alpha
success = 4,
failure = 50 - 4)[2],
c(0, 1e6)) # Search range for alpha

# Calculate the Bayes Factor and posterior probability using specific alpha and beta values.
# These specific values (alpha = 50/9, beta = 5*50/9 - 4) are likely the result of the
# optimization step above, or pre-determined values that are of interest.
p_post_null(1/6, 1/2, 50/9, 5 * 50/9 - 4, 4, 50 - 4)

http://rio2016.5ch.net/test/read.cgi/math/1723152147/856

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