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Re: Bernoulli variables again

• To: mathgroup at smc.vnet.net
• Subject: [mg46551] Re: Bernoulli variables again
• From: drbob at bigfoot.com (Bobby R. Treat)
• Date: Mon, 23 Feb 2004 02:15:42 -0500 (EST)
• References: <c1amhq$3qt$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

Because of the peculiar structure of the expression below, you may
profit from the transformation x[i_]->1-y[i]. A lot of extra
complexity is due to all those 1 - xi. If your problems can be cast in
terms of the y[i], this will help a lot. If you HAVE to go back to
x[i] at some point, I think I'd do it as late in the game as possible.
But I doubt it's necessary. After all, the y[i] are Bernoulli too,
with an easy transformation between the probabilities associated.

Bobby

jmyers6761 at aol.com (JMyers6761) wrote in message news:<c1amhq$3qt$1 at smc.vnet.net>...
> Consider the following expresion where each of the xn are Bernoulli variables:
> 
> 1 - (1 - (1 - (1 - x1) (1 - x2)) x4) (1 - (1 - (1 - x2) (1 - x3)) x5)
> 
> when this expression is Expanded we get:
> 
> x1 x4+x2 x4 - x1 x2 x4 + x2 x5 + x3 x5 - x2 x3 x5 - x1 x2 x4 x5 - x2^2 x4 x5 +
> x1 x2^2 x4 x5 - x1 x3 x4 x5 - x2 x3 x4 x5 + 2 x1 x2 x3 x4 x5 + x2^2 x3 x4 x5 -
> x1 x2^2 x3 x4 x5
> 
> but since a Bernoulli variable, x, can take on only values of 0 or 1 and x^n =
> x this expression is subject to the transformation x^n-> x with the following
> result:
> 
> x1 x4 + x2 x4 - x1 x2 x4 + x2 x5 + x3 x5 - x2 x3 x5 - x2 x4 x5 - x1 x3 x4 x5 +
> x1 x2 x3 x4 x5
> 
> My question is does anyone know how to transform such an unexpanded expression
> without having to first do an Expand? The actual expressions I am dealing with
> have a very large number of fully expanded terms ( > 10^6) and as a result
> Mathematica runs out of memory attemting to Expand the expressions. I know that
> the resulting expressions, after the Bernoulli transformation, if they could be
> expanded, would still be large but the number of terms would be much smaller,
> on the order of 10^3 terms. Does anyone know of a technique that could used to
> transform the unexpanded expression?
> 
> Thank you for your thoughts.
> 
> Al Myers