From 0e3311bf0601a8db66058439bd1fff2a9cac2e58 Mon Sep 17 00:00:00 2001
From: "Steinar H. Gunderson" <sgunderson@bigfoot.com>
Date: Tue, 20 Mar 2012 02:05:33 +0100
Subject: [PATCH] Invert the Hessian using Eigen instead of just using the
 diagonal. Gives slightly wider/better standard deviations, at the cost of
 more of numerical problems.

---
 bayeswf.cpp                | 49 ++++++++++++++------------------------
 www/ratings-explained.html |  2 +-
 2 files changed, 19 insertions(+), 32 deletions(-)

diff --git a/bayeswf.cpp b/bayeswf.cpp
index 71686c9..ca205ef 100644
--- a/bayeswf.cpp
+++ b/bayeswf.cpp
@@ -2,6 +2,8 @@
 #include <math.h>
 #include <string.h>
 #include <stdlib.h>
+#include <Eigen/Core>
+#include <Eigen/Eigenvalues>
 
 #include <map>
 #include <vector>
@@ -9,6 +11,7 @@
 #include <algorithm>
 
 using namespace std;
+using namespace Eigen;
 
 #define PRIOR_MU 1500
 #define PRIOR_WEIGHT 1.0
@@ -196,13 +199,13 @@ float compute_total_logl(float *mu, int num_players)
  *
  * Note that this does not depend on mu or the margin at all.
  */
-double hessian[MAX_PLAYERS][MAX_PLAYERS];
+double hessian[MAX_PLAYERS * MAX_PLAYERS];
 void construct_hessian(const float *mu, int num_players)
 {
 	memset(hessian, 0, sizeof(hessian));
 
 	for (int i = 0; i < num_players; ++i) {
-		hessian[i][i] += 1.0f / (prior_sigma * prior_sigma);
+		hessian[i * num_players + i] += 1.0f / (prior_sigma * prior_sigma);
 	}
 	for (unsigned i = 0; i < all_matches.size(); ++i) {
 		const match &m = all_matches[i];
@@ -213,44 +216,29 @@ void construct_hessian(const float *mu, int num_players)
 		double sigma_sq = global_sigma * global_sigma;
 		float w = m.weight;
 
-		hessian[p1][p2] -= w / sigma_sq;
-		hessian[p2][p1] -= w / sigma_sq;
+		hessian[p1 * num_players + p2] -= w / sigma_sq;
+		hessian[p2 * num_players + p1] -= w / sigma_sq;
 
-		hessian[p1][p1] += w / sigma_sq;
-		hessian[p2][p2] += w / sigma_sq;
+		hessian[p1 * num_players + p1] += w / sigma_sq;
+		hessian[p2 * num_players + p2] += w / sigma_sq;
 	}
 }
 
-// Compute uncertainty (stddev) of mu estimates, which is 1/sqrt(H_ii),
+// Compute uncertainty (stddev) of mu estimates, which is sqrt((H^-1)_ii),
 // where H is the Hessian (see construct_hessian()).
 void compute_mu_uncertainty(const float *mu, int num_players)
 {
-	memset(mu_stddev, 0, sizeof(mu_stddev));
-
-	// Temporarily use mu_stddev to store the diagonal of the Hessian.
-
-	// Prior.
+	Matrix<float, Dynamic, Dynamic> h(num_players, num_players);
 	for (int i = 0; i < num_players; ++i) {
-		mu_stddev[i] += 1.0f / (prior_sigma * prior_sigma);
-	}
-
-	// Matches.
-	for (unsigned i = 0; i < all_matches.size(); ++i) {
-		const match &m = all_matches[i];
-
-		int p1 = m.player;
-		int p2 = m.other_player;
-
-		double sigma_sq = global_sigma * global_sigma;
-		float w = m.weight;
-
-		mu_stddev[p1] += w / sigma_sq;
-		mu_stddev[p2] += w / sigma_sq;
+		for (int j = 0; j < num_players; ++j) {
+			h(i, j) = hessian[i * num_players + j];
+		}
 	}
 
-	// Now convert to standard deviation.
+	// FIXME: Use pseudoinverse if applicable.
+	Matrix<float, Dynamic, Dynamic> ih = h.inverse();
 	for (int i = 0; i < num_players; ++i) {
-		mu_stddev[i] = 1.0f / sqrt(mu_stddev[i]);
+		mu_stddev[i] = sqrt(ih(i, i));
 	}
 }
 
@@ -357,6 +345,7 @@ int main(int argc, char **argv)
 #if DUMP_RAW
 	dump_raw(mu, num_players);
 #else
+	construct_hessian(mu, num_players);
 	compute_mu_uncertainty(mu, num_players);
 	dump_scores(players, mu, mu_stddev, num_players);
 	//fprintf(stderr, "Optimal sigma: %f (two-player: %f)\n", sigma[0], sigma[0] * sqrt(2.0f));
@@ -365,7 +354,5 @@ int main(int argc, char **argv)
 
 	float total_logl = compute_total_logl(mu, num_players);
 	printf("%f 0 -4\n", total_logl);
-
-//	construct_hessian(mu, sigma, num_players);
 #endif
 }
diff --git a/www/ratings-explained.html b/www/ratings-explained.html
index c2e5c7f..c45075b 100755
--- a/www/ratings-explained.html
+++ b/www/ratings-explained.html
@@ -164,7 +164,7 @@
     <p>Dette var faktisk alt. Det skal sies at det sikkert er nok å ta tak i
       som ikke er blitt dekket her &ndash; for eksempel er det ikke beskrevet
       hvordan man regner ut <em>usikkerheten</em> i de estimerte ratingene
-      (hvilket er passe komplekst, og basert på en delvis invertering av
+      (hvilket er passe komplekst, og basert på å invertere
       <a href="http://en.wikipedia.org/wiki/Hessian_matrix">Hess-matrisen</a>
       til rimelighetsfunksjonen),
       eller hvordan modellen vekter kamper eldre kamper gis mindre betydning).</p>
-- 
2.39.2