高校数学の質問スレ（医者・東大卒専用） Part438 (991ﾚｽ)
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抽出解除 必死ﾁｪｯｶｰ(本家) (べ) 自ID ﾚｽ栞 あぼーん

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ﾘﾛｰﾄﾞ規制です｡10分ほどで解除するので､他のﾌﾞﾗｳｻﾞへ避難してください｡

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854: １３２人目の素数さん [sage] 05/30(金)06:24 ID:zoAuXcvc(1/3)
par(bty="l")
y = c(43, 24, 100, 35, 85)
yn = max(y)
n = length(y)
B = 200
like = numeric(B)
for(i in yn:B) { like[i] = 1 / (i^n) }
pmf = like / sum(like)
mean = sum((1:B) * pmf)
plot(pmf, main="Posterior PMF of N", xlab="N", ylab="Probability", type="h", col="blue")
plot(cumsum(pmf), main="Posterior CDF of N", xlab="N", ylab="Cumulative Probability", type="s", col="blue")
abline(h=0.95, lty=3)
c(lower=yn, mean=mean, upper=which(cumsum(pmf) > 0.95)[1])
sd = sqrt(sum(((1:B) - mean)^2 * pmf))
prob_N_gt_150 = sum(pmf[151:B])
cat("Posterior Mean:", mean, "\n")
cat("Posterior Standard Deviation:", sd, "\n")
cat("P(N > 150):", prob_N_gt_150, "\n")

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855: １３２人目の素数さん [sage] 05/30(金)19:44 ID:zoAuXcvc(2/3)
(* pbetat関数の定義 *)
pbetat[p0_, prob_, ab_, data_] := Module[{a, b, s, f, lbf, bf, post},
a = ab[[1]];
b = ab[[2]];
s = data[[1]];
f = data[[2]];
lbf = s * Log[p0] + f * Log[1 - p0] + Log@Beta[a, b] - Log@Beta[a + s, b + f];
bf = Exp[lbf];
post = prob * bf / (prob * bf + 1 - prob);
<|"bf" -> bf, "post" -> post|>
]

(* 関数fの定義 *)
f[alpha_] := pbetat[1/6, 0.5, {alpha, 5*alpha - 4}, {4, 50 - 4}]["post"]

(* 最小化 *)
result = NMinimize[{f[alpha], 0 <= alpha <= 20}, alpha]

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856: １３２人目の素数さん [sage] 05/30(金)20:13 ID:zoAuXcvc(3/3)
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)

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