Question

作为作业中的问题，我被要求编写一个Octave函数，模拟1000个实验，找到一个随机变量 X ，字母 {0,1,2,3} 和pmf：

Px（0）= 1/8

Px（1）= 1/4

Px（2）= 1/2

Px（3）= 1/8

通过询问一系列二元，“是”或“否”的问题。

我已经确定要求找到 X 的值的二进制问题的最佳序列是简单地询问“Is X = p？”其中p是可能的值，按概率递减的顺序排列。

所以最佳顺序是：

是X = 2？

如果不是：
是X = 1？

如果不是：
是X = 0？

如果没有，那么 X = 3

这是我写的功能：

function x = guessing_experiment(probabilities, n)
  % generates n simulations of finding a random number in an alphabet by asking binary questions,
  % where 'probabilities' is a list of the probabilities per number in the order the questions will be asked

  num_Qs = zeros(1,n);                            % allocate array of size n for number of questions asked per experiment
  [num_col, alphabet_size] = size(probabilities); % get size of alphabet

  for i = 1:n                                     % generate n experiments
    Qs = 0;                                       % number of questions asked in this experiment
    for j = 1:alphabet_size - 1                   % iterate through questions
      question = rand;                            % generate random number in range [0, 1]
      Qs++;                                       % incremenet number of questions asked
      if (question <= probabilities(j))           % if question produces a "yes" answer
        break;
      endif
    endfor
    num_Qs(i) = Qs;                               % store number of questions asked for this experiment
  endfor

  x = mean(num_Qs);                               % calculate mean number of questions asked over the n experiments 

 end

其中称为guessing_experiment([1/2, 1/4, 1/8, 1/8], 1000) 数组是每个问题产生“是”答案的概率，按照它们的询问顺序排列， n 是实验的数量。

提出这些问题应该会产生1.75的平均问题，但我的程序总是产生~1.87的平均值。我的脚本错误在哪里？

我假设它与生成一个新的随机数有关，以模拟所提出的3个问题中的每一个。

Answer 1

我删除了之前的错误答案，其中说明您的脚本是正确的，并且您的计算错误。我再次考虑它，你的计算是正确的。我自己尝试使用以下MATLAB脚本：

% probabilities for each number
p = [1/8,1/4,1/2,1/8];
% sort them from higher to lower
p = sort(p,'descend');
% number of questions per probability
nq = 1:length(p)-1;
% the last question can distinguish between two variables
nq(end+1) = nq(end);
% number of trials
n = 100000;
% random sample number of questions
q = randsample(nq,n,true,p);
% mean number of questions
avgQ = mean(q)

和获得的平均值。是1.75 - 正如你计算的那样。我将尝试再次查看您的代码以查看错误

修改

您的脚本存在的问题是您忽略了conditional probability，即在询问有关变量的问题时忽略了您已经了解的信息。例如，当您提出第三个问题时，该值0的概率不是 p=1/8而是p=1/2，因为您已经知道它不是1或2。您需要做的修复是将概率除以可能的事件概率probabilities(j)/sum(probabilities(j:end))：

n = 10000; p = [1/8,1/4,1/2,1/8]; % sort them from higher to lower probabilities = sort(p,'descend'); probabilities(end-1) = probabilities(end-1) + probabilities(end); probabilities(end) = []; alphabet_size = numel(probabilities); num_Qs = zeros(1,n); % allocate array of size n for number of questions asked per experiment for i = 1:n % generate n experiments Qs = 0; % number of questions asked in this experiment for j = 1:alphabet_size % iterate through questions question = rand; % generate random number in range [0, 1] Qs = Qs + 1; % incremenet number of questions asked if question < probabilities(j)/sum(probabilities(j:end)) % if question produces a "yes" answer break; end end num_Qs(i) = Qs; % store number of questions asked for this experiment end x = mean(num_Qs)

x~1.75

此场景中条件概率的向量是：

p = [1/8,1/4,1/2,1/8]; p = sort(p,'descend'); cond_p = p./cumsum(p,'reverse') cond_p = 0.5000 0.5000 0.5000 1.0000

通过询问二进制问题来模拟找到随机选择的数字

1 个答案: