Nested sampling manual - ประวัติรุ่นแก้ไข

Jung: /* $Next$ */

2007-04-10T12:32:54Z

Jung: /* References */

2007-04-10T12:21:40Z

References

Jung: /* $Next$ */

2007-04-10T12:19:47Z

Jung: /* Default values of parameters: the details */

2007-04-10T12:01:44Z

Default values of parameters: the details

Jung: /* Default values of parameters: the details */

2007-04-10T10:07:58Z

Default values of parameters: the details

Jung: /* Default values of parameters: the details */

2007-04-10T09:46:42Z

Default values of parameters: the details

Jung: /* Default values of parameters: the details */

2007-04-10T09:15:21Z

Default values of parameters: the details

Jung: /* $Ngp$ { if (''walk'' = 2) } */

2007-04-10T06:39:59Z

<math>Ngp</math> { if (''walk'' = 2) }

Jung: /* $Ngp$ \big(if (''walk'' = 2) \big) */

2007-04-10T06:39:02Z

<math>Ngp</math> \big(if (''walk'' = 2) \big)

Jung: /* Introduction */

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Introduction

@@ แถว 103: / แถว 103: @@
 \prod_{i=1}^I x_i \in \left[ \frac{m}{M} \div \exp \left(3\sqrt{\frac{\log \frac{M}{m}}{Next}} \right) , \frac{m}{M} \times \exp \left(3\sqrt{\frac{\log \frac{M}{m}}{Next}} \right) \right]
 </math>
 === <math>Nwalk</math> ===

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 12:21, 10 เมษายน 2550
แถว 113:		แถว 113:
	* Ramussen (2003). Bayesian Statistics 7.		* Ramussen (2003). Bayesian Statistics 7.
	* Gelman et al. (1996). Efficient Metropolis Jumping Rules. Bayesian Statistics 5.		* Gelman et al. (1996). Efficient Metropolis Jumping Rules. Bayesian Statistics 5.
		+	* Lovasz and Vempala (2004). Simulated Annealing in Convex Bodies and an O(n^4) Volume Algorithm. [http://citeseer.ist.psu.edu/673703.html link]

@@ แถว 89: / แถว 89: @@
 <math>
-Var[\sum_{i=1}^{I^*} \log x_i] = \sum_{i=1}^{I^*} Var[ \log x_i] = \frac{-I^*}{(Next)^2}.
+Var[\sum_{i=1}^{I^*} \log x_i] = \sum_{i=1}^{I^*} Var[ \log x_i] = \frac{I^*}{(Next)^2}.
 </math>

@@ แถว 73: / แถว 73: @@
 Niter \approx  Next \cdot  D \log D.
 </math>
 === <math>Next</math> ===
 Assume first that we fix <math> Next\, </math> to some positive integer.
-Let <math> I\, </math>  be the number of iterations needed to reach the fraction <math>m/M</math> for nested sampling (in previous subsection,  we try to approxmiate <math> I\, </math> by <math> Niter </math>). Let <math> x_i, for i = 1, ...,I\, </math> be a random variable of a ''shrinked mass'' for each iteration. Each
+Let <math> x_i \, </math> be a random variable of a ''shrinked mass'' for each iteration <math> i = 1, 2, ... </math>. According to nested sampling procedure, each <math> x_i \, </math> is an ''extreme value'' random variable with degree <math> Next</math>. Up to <math> I^{th}\, </math> iteration, the mass <math> M\, </math> is shrinked by <math> \prod_{i=1}^I x_i </math>. Taking logarithm, the log of remaining mass is <math> \log M + \sum_{i=1}^I \log x_i </math>.
 === <math>Nwalk</math> ===
-The default value of <math>D \log D</math> for <math>Nwalk</math> is adhoc. I try to make the total time complexity of nested sampling be comparable to that of MCMC methods for computing the volume of convex body, e.g. the work of Lovasz and Vempala (2003).
+The default value of <math>D \log D</math> for <math>Nwalk</math> is adhoc. I try to make the total time complexity of nested sampling be comparable to that of ''MCMC methods for computing the volume of convex body'', e.g. the work of Lovasz and Vempala (2003).
 ==References==

@@ แถว 56: / แถว 56: @@
 Let <math> M \, </math> be the mass of the <math>D\,</math>-dimensional parameter space and let <math> m \, </math> be a mass defined by a typical set with respect to a given posterior. Typically, when <math> D \, </math> increases, <math> m/M \, </math> converges to zero exponentially fast. Moreover, when we integrate over all parameter space with respect to the posterior, only this fraction of parameters contributes a value in the integral result.
-Thus, the iteration of nested sampling, <math> Niter \, </math> must be high enough for nested sampling to reach this region. Since for each iteration, the mass is reduce by the factor <math> \frac{Next}{Next + 1} \, </math>, we have
+Thus, the iteration of nested sampling, <math> Niter \, </math> must be high enough for nested sampling to reach this region. Since for each iteration, the mass '''on average''' is reduce by the factor <math> \frac{Next}{Next + 1} \, </math>, we have
 <math>
 \left(\frac{Next}{Next + 1} \right)^{Niter} = \frac{m}{M}.
 </math>
@@ แถว 67: / แถว 65: @@
 <math>
 Niter \approx  Next \cdot \log \frac {M}{m}.
 </math>
@@ แถว 75: / แถว 71: @@
 <math>
-Niter \approx  Next \cdot  D \log D.
+Let <math> I\, </math>  be the number of iterations needed to reach the fraction <math>m/M</math> for nested sampling (in previous subsection,  we try to approxmiate <math> I\, </math> by <math> Niter </math>). Let <math> x_i, for i = 1, ...,I\, </math> be a random variable of a ''shrinked mass'' for each iteration. Each
-</math>
+=== <math>Nwalk</math> ===
 ==References==

@@ แถว 53: / แถว 53: @@
 == Default values of parameters: the details ==
-Let <math> M /, </math> be the mass of the <math>D-/,</math>dimensional parameter space.
+===<math>Niter</math>===
 ==References==

@@ แถว 51: / แถว 51: @@
 ==== <math>Ngp</math>  { if (''walk'' = 2)  } (no default) ====
 we have to define this number. This defines the number of pseudo-walk using GP as an approximation of the real likelihood.
 ==References==

@@ แถว 49: / แถว 49: @@
 # walk = 2, use metropolis-hasting sampler with GP-proposal (our project)
-==== <math>Ngp</math>  { if (''walk'' = 2)  } ====
+==== <math>Ngp</math>  { if (''walk'' = 2)  } (no default) ====
 we have to define this number. This defines the number of pseudo-walk using GP as an approximation of the real likelihood.

@@ แถว 9: / แถว 9: @@
 ===NOTE for Oli===
 To test the correctness of our implementations, I also provide a simple MOGs likelihood
-for us [see (1.2) below]: a mixture of three spherical gaussian with <math> \sigma = 1 </math>, so that the
+for us [see (1.2) below]: a mixture of three spherical gaussian with <math> \sigma\, = 1 </math>, so that the
-integral result will be <math> 3 \cdot (2 \pi)^{D/2} </math> where <math>D</math> is the number of dimensions in the parameter space.
+integral result will be <math> 3 \cdot (2 \pi)^{D/2} </math> where <math>D\,</math> is the number of dimensions in the parameter space.
 ==Program parameters==

Nested sampling manual - ประวัติรุ่นแก้ไข

Jung: /* Next */

Jung: /* References */

Jung: /* Next */

Jung: /* Default values of parameters: the details */

Jung: /* Default values of parameters: the details */

Jung: /* Default values of parameters: the details */

Jung: /* Default values of parameters: the details */

Jung: /* Ngp { if (''walk'' = 2) } */

Jung: /* Ngp \big(if (''walk'' = 2) \big) */

Jung: /* Introduction */

Jung: /* $Next$ */

Jung: /* $Next$ */

Jung: /* $Ngp$ { if (''walk'' = 2) } */

Jung: /* $Ngp$ \big(if (''walk'' = 2) \big) */