From 974b4c027b9110b0efddfdd76afced5cd6df76c0 Mon Sep 17 00:00:00 2001 From: Chris Date: Thu, 9 Apr 2026 22:28:16 +0200 Subject: [PATCH] finished 3b --- Oblig/3b/3b.R | 171 ++++++++++ Oblig/3b/Chapter13Tasks13and14.png | Bin 0 -> 65699 bytes Oblig/3b/Chapter14Task12.png | Bin 0 -> 56297 bytes Oblig/3b/Chapter14Task14.png | Bin 0 -> 43284 bytes Oblig/3b/Chapter14Task16Part1.png | Bin 0 -> 44414 bytes Oblig/3b/Chapter14Task16Part2.png | Bin 0 -> 9700 bytes Oblig/3b/Oblig 3b - Inferens 2.pdf | Bin 0 -> 106522 bytes Oblig/3b/Oblig_3b.pdf | Bin 0 -> 312329 bytes Oblig/3b/oblig3b_writeup.tex | 486 +++++++++++++++++++++++++++++ 9 files changed, 657 insertions(+) create mode 100644 Oblig/3b/3b.R create mode 100644 Oblig/3b/Chapter13Tasks13and14.png create mode 100644 Oblig/3b/Chapter14Task12.png create mode 100644 Oblig/3b/Chapter14Task14.png create mode 100644 Oblig/3b/Chapter14Task16Part1.png create mode 100644 Oblig/3b/Chapter14Task16Part2.png create mode 100644 Oblig/3b/Oblig 3b - Inferens 2.pdf create mode 100644 Oblig/3b/Oblig_3b.pdf create mode 100644 Oblig/3b/oblig3b_writeup.tex diff --git a/Oblig/3b/3b.R b/Oblig/3b/3b.R new file mode 100644 index 0000000..4d13b1a --- /dev/null +++ b/Oblig/3b/3b.R @@ -0,0 +1,171 @@ +# Posterior for a Poisson process with Gamma(shape = k0, rate = t0) prior +# and data n events observed during time t. +poisson_posterior <- function(k0, t0, n, t) { + list(shape = k0 + n, rate = t0 + t) +} + +# Predictive probability for future counts in a Poisson process when +# lambda | data ~ Gamma(shape, rate). +# R's pnbinom uses the same negative-binomial form we need here. +predictive_count_cdf <- function(m, shape, rate, horizon) { + pnbinom(m, size = shape, prob = rate / (rate + horizon)) +} + +# Predictive probability for the k-th future waiting time. +# We use P(T_{+k} <= l) = P(N_{+l} >= k) = 1 - P(N_{+l} <= k - 1). +predictive_waiting_time_cdf <- function(k, l, shape, rate) { + 1 - predictive_count_cdf(k - 1, shape, rate, l) +} + +# Posterior probability P(X > Y) for independent gamma variables. +gamma_gt_prob <- function(shape_x, rate_x, shape_y, rate_y) { + integrate( + function(x) { + dgamma(x, shape = shape_x, rate = rate_x) * + pgamma(x, shape = shape_y, rate = rate_y) + }, + lower = 0, + upper = Inf, + subdivisions = 2000L, + rel.tol = 1e-10 + )$value +} + +# Exact probability P(U > V) for independent beta variables with integer +# first shape parameter in U, from the finite-sum identity used in Rule A.3.3. +beta_gt_prob_exact <- function(a, b, c, d) { + total <- 0 + for (i in 0:(a - 1)) { + total <- total + beta(c + i, b + d) / + ((b + i) * beta(1 + i, b) * beta(c, d)) + } + total +} + +# Normal approximation for pairwise beta comparison. +# If U and V are independent beta variables, then D = U - V is approximated +# by a normal variable with mean E(U) - E(V) and variance Var(U) + Var(V). +beta_compare_prob_normal <- function(a, b, c, d, direction = c("gt", "lt")) { + direction <- match.arg(direction) + + mean_u <- a / (a + b) + mean_v <- c / (c + d) + var_u <- a * b / ((a + b)^2 * (a + b + 1)) + var_v <- c * d / ((c + d)^2 * (c + d + 1)) + + mean_d <- mean_u - mean_v + sd_d <- sqrt(var_u + var_v) + + if (direction == "gt") { + 1 - pnorm(0, mean = mean_d, sd = sd_d) + } else { + pnorm(0, mean = mean_d, sd = sd_d) + } +} + +# Posterior probability P(mu_x > mu_y) for Gaussian samples under neutral priors. +# Marginally, each mean has a Student t posterior: +# mu | data ~ t_{n-1}(xbar, s / sqrt(n)). +gaussian_mean_gt_prob <- function(x, y) { + nx <- length(x) + ny <- length(y) + mx <- mean(x) + my <- mean(y) + sx <- sd(x) + sy <- sd(y) + + integrate( + function(z) { + dt((z - mx) / (sx / sqrt(nx)), df = nx - 1) / (sx / sqrt(nx)) * + pt((z - my) / (sy / sqrt(ny)), df = ny - 1) + }, + lower = -Inf, + upper = Inf, + subdivisions = 5000L, + rel.tol = 1e-10 + )$value +} + +# ----------------------------- +# Point 3a: Chapter 13, task 13d +# ----------------------------- + +post_13d <- poisson_posterior(k0 = 3, t0 = 73, n = 6, t = 119) + +cat("Chapter 13, task 13d\n") +cat("Posterior shape =", post_13d$shape, "\n") +cat("Posterior rate =", post_13d$rate, "\n\n") + +# ----------------------------- +# Point 3b: Chapter 13, task 14c +# ----------------------------- + +shape_14c <- 29 +rate_14c <- 8 +k_14c <- 3 +l_14c <- 1 +m_14c <- 5 + +p_n_le_m <- predictive_count_cdf(m = m_14c, shape = shape_14c, rate = rate_14c, horizon = l_14c) +p_t_le_l <- predictive_waiting_time_cdf(k = k_14c, l = l_14c, shape = shape_14c, rate = rate_14c) + +cat("Chapter 13, task 14c\n") +cat("P(N_{+1} <= 5) =", p_n_le_m, "\n") +cat("P(T_{+3} <= 1) =", p_t_le_l, "\n\n") + +# ----------------------------- +# Point 3c: Chapter 14, task 12 +# ----------------------------- + +odd <- c(22, 20, 21, 20, 21, 21, 19, 21) +kjell <- c(15, 12, 32, 12, 11, 13, 14) + +p_mu_odd_gt_kjell <- gaussian_mean_gt_prob(odd, kjell) + +cat("Chapter 14, task 12\n") +cat("Odd: n =", length(odd), "mean =", mean(odd), "sd =", sd(odd), "\n") +cat("Kjell: n =", length(kjell), "mean =", mean(kjell), "sd =", sd(kjell), "\n") +cat("P(mu_Odd > mu_Kjell | data) =", p_mu_odd_gt_kjell, "\n\n") + +# ----------------------------- +# Point 3d: Chapter 14, task 14 +# ----------------------------- + +p_14a <- gamma_gt_prob(7, 70, 4, 80) +p_14b <- 1 - gamma_gt_prob(9, 20, 11, 20) +p_14c <- 1 - gamma_gt_prob(90, 200, 110, 200) + +cat("Chapter 14, task 14\n") +cat("P(Theta > Psi) in 14a =", p_14a, "\n") +cat("P(Theta < Psi) in 14b =", p_14b, "\n") +cat("P(Theta < Psi) in 14c =", p_14c, "\n\n") + +# ----------------------------- +# Point 3e: Chapter 14, task 16 +# ----------------------------- + +# 16a: H1 is psi < pi, so exact probability is 1 - P(psi > pi). +p_16a_exact <- 1 - beta_gt_prob_exact(2, 5, 4, 3) +p_16a_norm <- beta_compare_prob_normal(2, 5, 4, 3, direction = "lt") + +# 16b: H1 is psi > pi. +p_16b_exact <- beta_gt_prob_exact(23, 17, 17, 23) +p_16b_norm <- beta_compare_prob_normal(23, 17, 17, 23, direction = "gt") + +# 16c: H1 is psi > pi. +p_16c_exact <- beta_gt_prob_exact(20, 20, 17, 23) +p_16c_norm <- beta_compare_prob_normal(20, 20, 17, 23, direction = "gt") + +# 16d: same as 16c, but all parameters multiplied by 10. +p_16d_exact <- beta_gt_prob_exact(200, 200, 170, 230) +p_16d_norm <- beta_compare_prob_normal(200, 200, 170, 230, direction = "gt") + +cat("Chapter 14, task 16\n") +cat("16a exact =", p_16a_exact, "\n") +cat("16a normal =", p_16a_norm, "\n") +cat("16b exact =", p_16b_exact, "\n") +cat("16b normal =", p_16b_norm, "\n") +cat("16c exact =", p_16c_exact, "\n") +cat("16c normal =", p_16c_norm, "\n") +cat("16d exact =", p_16d_exact, "\n") +cat("16d normal =", p_16d_norm, "\n") diff --git a/Oblig/3b/Chapter13Tasks13and14.png b/Oblig/3b/Chapter13Tasks13and14.png new file mode 100644 index 0000000000000000000000000000000000000000..9bd7cb36853e88ddfdede84c7b7f7f288204f7cc GIT binary patch literal 65699 zcmZU*2RN2({5O6{Av>v185KgLk}YM+sBGCQd#^$$D>5pXkxIxG84;qaNKqjx60$2J 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z%@C{EfEVsdtU3<&FbfKuKo4h~%i0oYMLidj1Ke*Lq{j?(t>%{vzPHqVH%r2|0x-F6 zX3+HFr@1N4{I)-?;SFGH-_FP$0?$ASdn%>B{4U^%PqA$3NpO<&bl#syw(qSC+C| zpnUtMm8FumjKy{DR^lC3xVzLd()2xcB1?I$W&1{)W~(Z4IoPFf?s~QB6AjNby~aEb z;HHv8hyD7Q?Y}BZ8wWsTDf^Gg((1gY@2|>|dExMC08m+qAMY>RW4COS&6?2DdnBh& zT=D&m7krfyvQo>=F877xx?bMT2s4Sc2jfjk}0Q9dIDMgHGU2L z=__;O*`yj*oZt=qNgP+H%s z+_6Z{E+QS&);YweM>j+^js>we#yR4_tw#vOU)1X$l2zX=GZ?s*5nFA0mU21?ng(Y& z;B01T*}C8;ZGpJ4ZetOM{jP1oXZO15I2PglzpDCQS5+DMf2wL^Prdc+g5z*;-=*aA zKU8%9yXf*4sQ$TE<|@Y1(ba=rRE%Fr4QNF4yyxn24d^!H7rmw^BQ2vKEhh~F%Yh+a zu=zjMI9*%E));BZ?{yDoM0E6VLwa~19e~B!NFOg(4}0{lXu7WU|31)7V|7zW4UDt% zU0W9ypv6(t*zq2kAGo{OI{=Zz`EMgV(7;sqWhH^F%fn<8fKvixBo!pUl5*0LG7x?X z(bEa&+^HxZsUgzgpFNpi zsO-OK+WftF^QNuG12q3>XQ!I0Gv@ zff?TOMe@r-_@!=Knx=ozS83;DFXjDseJlv*FN=A)6;dGzvKv m$w|~n`KSK#H=6%{;0 \mu_K, +\] +med signifikans $\alpha=0.2$. I denne typen oppgave velger vi $H_1$ dersom +\[ +P(\mu_O > \mu_K \mid \text{data}) > 1-\alpha = 0.8. +\] + +Ved numerisk integrasjon far vi +\[ +P(\mu_O > \mu_K \mid \text{data}) \approx 0.9392. +\] +Siden $0.9392 > 0.8$, velger vi $H_1$. Dataene gir derfor tilstrekkelig stotte for at +Odds hosteanfall i gjennomsnitt varer lenger enn Kjells. + +\subsection{Kapittel 14, oppgave 14} +Vi bruker samme beslutningsregel gjennom hele oppgaven: velg $H_1$ nar den posterior +sannsynligheten for utsagnet i $H_1$ er storre enn $1-\alpha$. + +\paragraph{14a} +Vi har +\[ +\Theta \sim \Gamma(7,70), +\qquad +\Psi \sim \Gamma(4,80), +\] +og hypotesen +\[ +H_1:\Theta > \Psi, +\qquad \alpha = 0.05. +\] +Den relevante posterior sannsynligheten er +\[ +P(\Theta > \Psi) += +\int_0^\infty f_\Theta(x)F_\Psi(x)\,dx +\approx 0.8774. +\] +Siden $0.8774 < 0.95$, velger vi $H_0$. + +\paragraph{14b} +Vi har +\[ +\Theta \sim \Gamma(9,20), +\qquad +\Psi \sim \Gamma(11,20), +\] +og hypotesen +\[ +H_1:\Theta < \Psi, +\qquad \alpha = 0.1. +\] +Her gir beregningen +\[ +P(\Theta < \Psi)\approx 0.6762. +\] +Siden $0.6762 < 0.9$, velger vi $H_0$. + +\paragraph{14c} +Nar parameterne i 14b ganges med $10$, far vi +\[ +\Theta \sim \Gamma(90,200), +\qquad +\Psi \sim \Gamma(110,200), +\] +med samme hypotese som i 14b. Da blir +\[ +P(\Theta < \Psi)\approx 0.9220. +\] +Siden $0.9220 > 0.9$, velger vi na $H_1$. + +Det er altsa en forskjell: med ti ganger sa mye informasjon blir posteriorfordelingene +mer konsentrerte, og det blir lettere a skille mellom parameterne. + +\subsection{Kapittel 14, oppgave 16} +Her sammenligner vi beta-fordelte variable. Jeg viser bade eksakt beregning og +normalapproksimasjon. + +For normalapproksimasjonen bruker vi at dersom +\[ +U \sim \beta(a,b), +\] +sa er +\[ +E(U)=\frac{a}{a+b}, +\qquad +\operatorname{Var}(U)=\frac{ab}{(a+b)^2(a+b+1)}. +\] +For to uavhengige variable $\psi$ og $\pi$ approksimerer vi deretter +\[ +D=\psi-\pi \approx N\!\bigl(E(\psi)-E(\pi),\operatorname{Var}(\psi)+\operatorname{Var}(\pi)\bigr). +\] + +\paragraph{16a} +\[ +\psi \sim \beta(2,5), +\qquad +\pi \sim \beta(4,3), +\qquad +H_1:\psi < \pi, +\qquad +\alpha=0.1. +\] +Eksakt beregning gir +\[ +P(\psi < \pi) \approx 0.8788, +\] +mens normalapproksimasjonen gir +\[ +P(\psi < \pi) \approx 0.8861. +\] +Begge er mindre enn $0.9$, sa vi velger $H_0$. + +\paragraph{16b} +\[ +\psi \sim \beta(23,17), +\qquad +\pi \sim \beta(17,23), +\qquad +H_1:\psi > \pi, +\qquad +\alpha=0.1. +\] +Eksakt beregning gir +\[ +P(\psi > \pi) \approx 0.9131, +\] +og normalapproksimasjonen gir +\[ +P(\psi > \pi) \approx 0.9153. +\] +Begge er storre enn $0.9$, sa vi velger $H_1$. + +\paragraph{16c} +\[ +\psi \sim \beta(20,20), +\qquad +\pi \sim \beta(17,23), +\qquad +H_1:\psi > \pi, +\qquad +\alpha=0.05. +\] +Eksakt beregning gir +\[ +P(\psi > \pi) \approx 0.7520, +\] +og normalapproksimasjonen gir +\[ +P(\psi > \pi) \approx 0.7527. +\] +Begge er mindre enn $0.95$, sa vi velger $H_0$. + +\paragraph{16d} +Nar parameterne i 16c ganges med $10$, far vi +\[ +\psi \sim \beta(200,200), +\qquad +\pi \sim \beta(170,230), +\qquad +H_1:\psi > \pi, +\qquad +\alpha=0.05. +\] +Eksakt beregning gir +\[ +P(\psi > \pi) \approx 0.9834, +\] +og normalapproksimasjonen gir +\[ +P(\psi > \pi) \approx 0.9837. +\] +Begge er storre enn $0.95$, sa vi velger $H_1$. + +Dette viser at mer data kan gi en tydelig forskjell selv nar forholdet mellom positive og +negative observasjoner er det samme som for 16c. + +\subsection{R-kode} +Listing~\ref{lst:oblig3b-r} viser R-koden som ble brukt til beregningene. + +\begin{listing}[H] +\begin{minted}[fontsize=\small, linenos, breaklines, frame=lines]{r} +# Oblig 3b calculations for points 1 and 3. +# This script uses only ASCII characters. + +# ----------------------------- +# Helper functions +# ----------------------------- + +# Posterior for a Poisson process with Gamma(shape = k0, rate = t0) prior +# and data n events observed during time t. +poisson_posterior <- function(k0, t0, n, t) { + list(shape = k0 + n, rate = t0 + t) +} + +# Predictive probability for future counts in a Poisson process when +# lambda | data ~ Gamma(shape, rate). +# R's pnbinom uses the same negative-binomial form we need here. +predictive_count_cdf <- function(m, shape, rate, horizon) { + pnbinom(m, size = shape, prob = rate / (rate + horizon)) +} + +# Predictive probability for the k-th future waiting time. +# We use P(T_{+k} <= l) = P(N_{+l} >= k) = 1 - P(N_{+l} <= k - 1). +predictive_waiting_time_cdf <- function(k, l, shape, rate) { + 1 - predictive_count_cdf(k - 1, shape, rate, l) +} + +# Posterior probability P(X > Y) for independent gamma variables. +gamma_gt_prob <- function(shape_x, rate_x, shape_y, rate_y) { + integrate( + function(x) { + dgamma(x, shape = shape_x, rate = rate_x) * + pgamma(x, shape = shape_y, rate = rate_y) + }, + lower = 0, + upper = Inf, + subdivisions = 2000L, + rel.tol = 1e-10 + )$value +} + +# Exact probability P(U > V) for independent beta variables with integer +# first shape parameter in U, from the finite-sum identity used in Rule A.3.3. +beta_gt_prob_exact <- function(a, b, c, d) { + total <- 0 + for (i in 0:(a - 1)) { + total <- total + beta(c + i, b + d) / + ((b + i) * beta(1 + i, b) * beta(c, d)) + } + total +} + +# Normal approximation for pairwise beta comparison. +# If U and V are independent beta variables, then D = U - V is approximated +# by a normal variable with mean E(U) - E(V) and variance Var(U) + Var(V). +beta_compare_prob_normal <- function(a, b, c, d, direction = c("gt", "lt")) { + direction <- match.arg(direction) + + mean_u <- a / (a + b) + mean_v <- c / (c + d) + var_u <- a * b / ((a + b)^2 * (a + b + 1)) + var_v <- c * d / ((c + d)^2 * (c + d + 1)) + + mean_d <- mean_u - mean_v + sd_d <- sqrt(var_u + var_v) + + if (direction == "gt") { + 1 - pnorm(0, mean = mean_d, sd = sd_d) + } else { + pnorm(0, mean = mean_d, sd = sd_d) + } +} + +# Posterior probability P(mu_x > mu_y) for Gaussian samples under neutral priors. +# Marginally, each mean has a Student t posterior: +# mu | data ~ t_{n-1}(xbar, s / sqrt(n)). +gaussian_mean_gt_prob <- function(x, y) { + nx <- length(x) + ny <- length(y) + mx <- mean(x) + my <- mean(y) + sx <- sd(x) + sy <- sd(y) + + integrate( + function(z) { + dt((z - mx) / (sx / sqrt(nx)), df = nx - 1) / (sx / sqrt(nx)) * + pt((z - my) / (sy / sqrt(ny)), df = ny - 1) + }, + lower = -Inf, + upper = Inf, + subdivisions = 5000L, + rel.tol = 1e-10 + )$value +} + +# ----------------------------- +# Point 3a: Chapter 13, task 13d +# ----------------------------- + +post_13d <- poisson_posterior(k0 = 3, t0 = 73, n = 6, t = 119) + +cat("Chapter 13, task 13d\n") +cat("Posterior shape =", post_13d$shape, "\n") +cat("Posterior rate =", post_13d$rate, "\n\n") + +# ----------------------------- +# Point 3b: Chapter 13, task 14c +# ----------------------------- + +shape_14c <- 29 +rate_14c <- 8 +k_14c <- 3 +l_14c <- 1 +m_14c <- 5 + +p_n_le_m <- predictive_count_cdf(m = m_14c, shape = shape_14c, rate = rate_14c, horizon = l_14c) +p_t_le_l <- predictive_waiting_time_cdf(k = k_14c, l = l_14c, shape = shape_14c, rate = rate_14c) + +cat("Chapter 13, task 14c\n") +cat("P(N_{+1} <= 5) =", p_n_le_m, "\n") +cat("P(T_{+3} <= 1) =", p_t_le_l, "\n\n") + +# ----------------------------- +# Point 3c: Chapter 14, task 12 +# ----------------------------- + +odd <- c(22, 20, 21, 20, 21, 21, 19, 21) +kjell <- c(15, 12, 32, 12, 11, 13, 14) + +p_mu_odd_gt_kjell <- gaussian_mean_gt_prob(odd, kjell) + +cat("Chapter 14, task 12\n") +cat("Odd: n =", length(odd), "mean =", mean(odd), "sd =", sd(odd), "\n") +cat("Kjell: n =", length(kjell), "mean =", mean(kjell), "sd =", sd(kjell), "\n") +cat("P(mu_Odd > mu_Kjell | data) =", p_mu_odd_gt_kjell, "\n\n") + +# ----------------------------- +# Point 3d: Chapter 14, task 14 +# ----------------------------- + +p_14a <- gamma_gt_prob(7, 70, 4, 80) +p_14b <- 1 - gamma_gt_prob(9, 20, 11, 20) +p_14c <- 1 - gamma_gt_prob(90, 200, 110, 200) + +cat("Chapter 14, task 14\n") +cat("P(Theta > Psi) in 14a =", p_14a, "\n") +cat("P(Theta < Psi) in 14b =", p_14b, "\n") +cat("P(Theta < Psi) in 14c =", p_14c, "\n\n") + +# ----------------------------- +# Point 3e: Chapter 14, task 16 +# ----------------------------- + +# 16a: H1 is psi < pi, so exact probability is 1 - P(psi > pi). +p_16a_exact <- 1 - beta_gt_prob_exact(2, 5, 4, 3) +p_16a_norm <- beta_compare_prob_normal(2, 5, 4, 3, direction = "lt") + +# 16b: H1 is psi > pi. +p_16b_exact <- beta_gt_prob_exact(23, 17, 17, 23) +p_16b_norm <- beta_compare_prob_normal(23, 17, 17, 23, direction = "gt") + +# 16c: H1 is psi > pi. +p_16c_exact <- beta_gt_prob_exact(20, 20, 17, 23) +p_16c_norm <- beta_compare_prob_normal(20, 20, 17, 23, direction = "gt") + +# 16d: same as 16c, but all parameters multiplied by 10. +p_16d_exact <- beta_gt_prob_exact(200, 200, 170, 230) +p_16d_norm <- beta_compare_prob_normal(200, 200, 170, 230, direction = "gt") + +cat("Chapter 14, task 16\n") +cat("16a exact =", p_16a_exact, "\n") +cat("16a normal =", p_16a_norm, "\n") +cat("16b exact =", p_16b_exact, "\n") +cat("16b normal =", p_16b_norm, "\n") +cat("16c exact =", p_16c_exact, "\n") +cat("16c normal =", p_16c_norm, "\n") +cat("16d exact =", p_16d_exact, "\n") +cat("16d normal =", p_16d_norm, "\n") +\end{minted} +\caption{R-script for beregningene i Oblig 3b} +\label{lst:oblig3b-r} +\end{listing}