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Problem: Consider a simple coin-flipping experiment in which we are given a pair of coins A and B of unknown biases, $\theta_{A}$ and $\theta_{B}$ respectively (that is, on any given flip, coin A will land on heads with probability $\theta_{A}$ and tails with probability $1-\theta_{A}$ and similar for coin B).

Repeating the following procedure five times: randomly choose one of the two coins, and perform 10 independent coin tosses with the selected coin. The probability of selecting coin A and coin B for each set is equal and this selection is made once per set of 10 tosses.

Suppose that we keep track of two vectors $x=(x_{1},x_{2},\ldots,x_{5})$ and $z=(z_{1},z_{2},\ldots,z_{5})$, where $x_{i}\in\{0,1,\ldots,10\}$ is the number of heads observed during the $i^{th}$ set of tosses, and $z_{i}\in\{A,B\}$ is the identity of the coin used during the $i^{th}$ set of tosses. Then the outcomes of 5 sets, can be represented by a vector $y=(y_{1},y_{2},y_{3},y_{4},y_{5})$. Each element in vector y, is denoted by $y_{i}=(x_{i},z_{i})$.

For example, $y_{3}=(x_{3}=8,z_{3}=A)$ means that there are 8 heads among 10 tosses for the 3rd set given coin A is chosen.

Let $f(y|\theta)$ denote the probability density function (PDF) that specifies the probability of observing data vector $y=(y_{1},y_{2},\ldots,y_{i},\ldots,y_{5})$ given the parameter $\theta=(\theta_{A},\theta_{B})$. If individual observation, $y_{i}$'s are statistically independent of one another, then according to the theory of probability, the PDF for the observation data $y=(y_{1},y_{2},\ldots,y_{i},\ldots,y_{5})$ given the parameter vector $\theta=(\theta_{A},\theta_{B})$ can be expressed as a multiplication of PDFs for individual observations

\begin{equation} \label{1} f(y=(y_{1},y_{2},\ldots,y_{i},\ldots,y_{5})|\theta)=f_{1}(y_{1}|\theta)f_{2}(y_{2}|\theta)\ldots f_{5}(y_{5}|\theta) \end{equation}

In a simplest case, there are only one observation and one parameter, that is, $y$ is a single element variable, and $\theta$ is a single element constant. Suppose that the single element independent variable y represents the number of heads observed in a single set of 10 Bernoulli trials(e.g. tossing a coin 10 times) and that the probability of a head on any one trial, represented by the single element parameter $\theta$; is 0.8. The PDF in this case is given by \begin{equation} f(y|n=10,\theta=0.8)=\frac{10!}{y!(10-y)!}(0.8)^{y}(0.2)^{10-y} \end{equation} where (y=0,1,...,10).

In the case I described at the beginning, there are 5 observations and two parameters, that is, the independent variable of y is a 5-element variable here $y=(y_{1},y_{2},\ldots,y_{i},\ldots,y_{5})$ and the parameter $\theta$ is a two element vector $\theta=(\theta_{A},\theta_{B})$ given that $\theta_{A}=0.8$, $\theta_{B}=0.45$.

Q: what is equation of PDF of $f(y=(y_{1},y_{2},\ldots,y_{i},\ldots,y_{5})| \theta=(\theta_{A}=0.8,\theta_{B}=0.45))$ here?

The reason I want to ask this question is to find out the maximum likelihood function since MLE is a inverse problem here.

Thanks

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1 Answer 1

There are three problems in your question that I think need to be addressed before it can be meaningfully answered.

  • The specification $n=50$ appears out of nowhere at the end; you had neither introduced a variable $n$ for the number of tosses, nor indicated that it should be considered as a parameter and not fixed.

  • You don't specify the probabilities for coins $A$ and $B$ to be used for any of the $5$ sets of tosses. Perhaps you're assuming that they're selected with equal probabilities as in the tutorial you'd asked about in an earlier question.

  • You introduce $x$ and $z$ as vectors of five variables each recording the number of heads and the coin used, respectively, in the $i$-th set. Then you write that each $y_i=(x,z)$. That doesn't make any sense. There's a free index $i$ on the left-hand side but not on the right-hand side; but even if we drop the index and assume that you meant $y=(x,z)$, it still doesn't make any sense, since there's a vector of $50$ individual tosses on the left and a vector of aggregated data containing a total of $10$ entries on the right.

[Edit in response to the edit in the question]

The question is still unclear on all three points.

  • You've now introduced $n=50$ before using it, but it's still not clear what it's doing in the condition part of a conditional probability. This would seem to indicate that you're considering other possible values of $n$ and the value of $n$ is determined randomly; that seems to be inconsistent with the rest of your description, which assumed $n$ fixed all along.
  • I only understand "each coin is selected with equal probability" by consulting the other questions and the tutorial; to make the question self-contained, you'd need to specify that this choice is made once per set of $10$ tosses.
  • The relationship between $y$ and $(x,z)$ isn't any clearer than before. $f(x,z|\theta)$ cannot be the PDF for $y$ because $x$ and $z$ contain less information than $y$; the results for sets of $10$ tosses each are aggregated in $x$, whereas they aren't aggregated in $y$.

[Edit in response to the second edit:]

It's now slightly less difficult to guess what you're trying to get at, but your notation still needs considerable work. You don't explain the meaning of the additional indices $j$ and $k$, and I get the impression that what you in fact mean is $y_i=(x_i,z_i)$. (Also note that the $y_i$ are now defined quite differently from the version in the original question; it would help to either delete the old definition or introduce a new symbol for this new quantity instead.)

You still haven't addresses what the $n=10$ is doing in the condition part of a conditional probability – nothing else in the entire post suggests that this is an unknown parameter or a random variable.

Also, $50!$ in the numerator of your function $f$ should be $10$, and you've swapped the probabilities $\theta$ and $1-\theta$ of success and failure.

However, since I think I can now guess what you're trying to ask, here's an answer to that:

$$p(x,z)=\frac1{2^5}\prod_{i=1}^5\binom{10}{x_i}\theta_{z_i}^{x_i}(1-\theta_{z_i})^{10-x_i}\;.$$

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@user19262: I've responded to your second edit. –  joriki Nov 17 '11 at 13:09
    
Thanks for you answer. @joriki: Taking your advice, I have edited the post (redefine the notations that confuse you, and delete the old post). Can you take a look at this updated post and see it is easy to read now. I can edit it again if it is still confusing. Can you also briefly explain how the PDF equation you gave to me is derived here? Thanks. I could understand the product part which is product of 5 Bernoulli distribution, but I could not figure out that (1/(2^5)) at front. –  xuesong Nov 19 '11 at 14:14

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