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I'm a bit of a novice at maths and want to learn more about algebraically manipulating likelihoods in statistics. There are a lot of equations that involve taking the product of a set of values given a model.

I know a few rules for manipulating sigma-notation (e.g., here and here).

  • What are the basic rules for manipulating sequences of products (i.e, $\prod_{i=1}^{I} ... $)? Is there a web page that you could direct me to?

e.g.,

  • $\prod_{i=1}^{I} x_i$
  • $\prod_{i=1}^{I} x_i y_i$
  • $\prod_{i=1}^{I} a + b x_i$
  • $\prod_{i=1}^{I} \exp x_i$
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1  
As one of my professors once said: "Products are bad. Take logs." Log turns the product into a sum. In the context of likelihoods, this usually works pretty well. –  Ted Dec 9 '11 at 5:14
    
@Ted thanks. I realize that it's simpler to maximize the log likelihood. However, I'd still like to consolidate my skills in manipulating products. –  mycat Dec 9 '11 at 5:18

3 Answers 3

up vote 4 down vote accepted

You just need to think about what the product notation means, and you can work out the rules yourself.

Take one of your examples:

$$\prod_{i=1}^I x_i y_i = \left(\prod_{i=1}^I x_i\right)\left(\prod_{i=1}^I y_i\right)$$

This rule works because the left side multiplies $x_1$ times $y_1$ times $x_2$ times $y_2$ etc., and the right side multiplies the x's first, then multiplies the result by the y's. Clearly those are equal.

Another one of your examples:

$$\prod_{i=1}^I \exp x_i = \exp \left( \sum_{i=1}^I x_i \right)$$

Can you figure out why this rule works?

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Thanks. Perhaps working them out myself is the best option. I can see that $e^{x_1} e^{x_2} = e^{x_1 + x_2}$, so that all makes sense. However, I'd still like to read a simple overview of all these kinds of operations, including the examples that I couldn't think of. When I get enough rep, I'll +1 you. –  mycat Dec 9 '11 at 5:29
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Another useful one is that powers distribute over products: $(\prod x_i)^a = \prod (x_i^a)$, which is the analog of this rule for sums: $a \sum (x_i) = \sum (a x_i)$. –  Ted Dec 9 '11 at 5:33

Whenever I think of pi-notation, I think of smaller finite products. It's not perfect - one has to worry about convergence problems when the products are infinite, but it's a pretty good tool to get an idea of what's possible and not.

For all of the questions you ask about, for instance, just thinking of 2 or 3 terms suffices.

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This might be inappropriate for an answer but I believe you tried yourself too hard at here. The $\prod $ sign just means multiplying some elements together, with a label in the bottom to denote the beginning element and a label on the top to denote the end element. The files you provided are very good and you should get a sense of what the general situation is. You will get better in using it if you read articles proving things using this symbol. But as any other short hand symbol writing the product of elements in this way does not really simplify anything; you should be able to write out any such products explicitly if you can write it in the $\prod$ sign.

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