Estimating joint and conditional probabilities with incomplete information - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T03:23:57Z http://mathoverflow.net/feeds/question/98300 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/98300/estimating-joint-and-conditional-probabilities-with-incomplete-information Estimating joint and conditional probabilities with incomplete information tvladeck 2012-05-29T19:04:18Z 2012-06-12T19:02:06Z <p>I'm working on an application for which it would be great to have the following functionality: </p> <p>Say that you have a collection $C$ of $n$ events, for now let's set $n = 3$ and call the events $a, b,$ and $c$</p> <p>Given $k$, say that for any subset $K \subset C$ s.t. $||K|| = k &lt; n$ of the set of events, you can produce the complete set of joint and conditional probabilities for any combination of the $k$ events. So, if $n = 3$ and $k = 2$, you would have complete information about</p> <p>$p(a | b)$, $p(b | a)$, $p(a | \neg b)$, $p(a \wedge b)$, $p(\neg (a \wedge b))$, etc., etc.</p> <p>$p(a | c)$, $p(c | a)$, $p(a | \neg c)$, $p(a \wedge c)$, $p(\neg (a \wedge c))$, etc., etc.</p> <p>$p(c | b)$, $p(b | c)$, $p(c | \neg b)$, $p(c \wedge b)$, $p(\neg (c \wedge b))$, etc., etc.</p> <p>Is there a 'correct' way to estimate the corresponding values for the joint and conditional probabilities for a, b, AND c? </p> <p>Things such as $p(a \wedge b \wedge c)$, $p(\neg (a \wedge b \wedge c))$, $p(a | b \wedge c)$, $p(a \wedge b | c)$, etc., etc.</p> <p>I am especially interested in the general case. </p> <h1>Edit</h1> <p>I should have been more expansive in my questions, as regardless of whether or not there is a 'correct' way of answering this question, if there are methods that allow one to say anything interesting about the bounds, or anything interesting at all, really, about the joint &amp; conditional probabilities of a, b, and c, than those methods will be very useful to me. </p> http://mathoverflow.net/questions/98300/estimating-joint-and-conditional-probabilities-with-incomplete-information/98311#98311 Answer by mike for Estimating joint and conditional probabilities with incomplete information mike 2012-05-29T21:20:53Z 2012-05-29T21:20:53Z <p>No, and to see that there is not, you might produce a collection of rademacher ($\pm 1$) random variables that are pairwise independent but not independent. No statement about 2 at a time, etc., would be different that for independent rademacher , but statements about the 3 way etc would be. You can do this, and generalise it to an N-K setting , by taking $X_1,X_2,X_3$ i.i.d. and $= \pm 1$ with prob $\frac 12$, and $Z_1 = X_2X_3, Z_2 = X_1X_3, Z_3 = X_1X_2$ .</p> http://mathoverflow.net/questions/98300/estimating-joint-and-conditional-probabilities-with-incomplete-information/99384#99384 Answer by R Hahn for Estimating joint and conditional probabilities with incomplete information R Hahn 2012-06-12T19:02:06Z 2012-06-12T19:02:06Z <p>This paper addresses a similar problem I think, although I believe they consider binary outcomes only: </p> <p><a href="http://uai.sis.pitt.edu/papers/07/p310-ramsahai.pdf" rel="nofollow">Ramsahai, R.R. (2007). Causal bounds and instruments. In Proceedings of the 23rd Annual Conference on Uncertainty in Artifical Intelligence, 310-317.</a></p> <p>The main result is a way to produce the sorts of upper and lower probabilities that Douglas Zare mentions in his comment. They additionally note a freely available software package that they use, called polymake.</p>