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1 <html>
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2 <head>
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3 <title>
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4 Netlab Reference Manual hmc
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5 </title>
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6 </head>
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7 <body>
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8 <H1> hmc
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9 </H1>
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10 <h2>
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11 Purpose
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12 </h2>
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13 Hybrid Monte Carlo sampling.
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14
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15 <p><h2>
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16 Synopsis
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17 </h2>
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18 <PRE>
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19
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20 samples = hmc(f, x, options, gradf)
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21 samples = hmc(f, x, options, gradf, P1, P2, ...)
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22 [samples, energies, diagn] = hmc(f, x, options, gradf)
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23 s = hmc('state')
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24 hmc('state', s)
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25 </PRE>
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26
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27
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28 <p><h2>
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29 Description
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30 </h2>
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31 <CODE>samples = hmc(f, x, options, gradf)</CODE> uses a
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32 hybrid Monte Carlo algorithm to sample from the distribution <CODE>p ~ exp(-f)</CODE>,
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33 where <CODE>f</CODE> is the first argument to <CODE>hmc</CODE>.
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34 The Markov chain starts at the point <CODE>x</CODE>, and the function <CODE>gradf</CODE>
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35 is the gradient of the `energy' function <CODE>f</CODE>.
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36
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37 <p><CODE>hmc(f, x, options, gradf, p1, p2, ...)</CODE> allows
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38 additional arguments to be passed to <CODE>f()</CODE> and <CODE>gradf()</CODE>.
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39
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40 <p><CODE>[samples, energies, diagn] = hmc(f, x, options, gradf)</CODE> also returns
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41 a log of the energy values (i.e. negative log probabilities) for the
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42 samples in <CODE>energies</CODE> and <CODE>diagn</CODE>, a structure containing
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43 diagnostic information (position, momentum and
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44 acceptance threshold) for each step of the chain in <CODE>diagn.pos</CODE>,
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45 <CODE>diagn.mom</CODE> and
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46 <CODE>diagn.acc</CODE> respectively. All candidate states (including rejected ones)
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47 are stored in <CODE>diagn.pos</CODE>.
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48
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49 <p><CODE>[samples, energies, diagn] = hmc(f, x, options, gradf)</CODE> also returns the
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50 <CODE>energies</CODE> (i.e. negative log probabilities) corresponding to the samples.
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51 The <CODE>diagn</CODE> structure contains three fields:
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52
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53 <p><CODE>pos</CODE> the position vectors of the dynamic process.
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54
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55 <p><CODE>mom</CODE> the momentum vectors of the dynamic process.
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56
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57 <p><CODE>acc</CODE> the acceptance thresholds.
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58
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59 <p><CODE>s = hmc('state')</CODE> returns a state structure that contains the state of the
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60 two random number generators <CODE>rand</CODE> and <CODE>randn</CODE> and the momentum of
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61 the dynamic process. These are contained in fields
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62 <CODE>randstate</CODE>, <CODE>randnstate</CODE>
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63 and <CODE>mom</CODE> respectively. The momentum state is
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64 only used for a persistent momentum update.
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65
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66 <p><CODE>hmc('state', s)</CODE> resets the state to <CODE>s</CODE>. If <CODE>s</CODE> is an integer,
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67 then it is passed to <CODE>rand</CODE> and <CODE>randn</CODE> and the momentum variable
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68 is randomised. If <CODE>s</CODE> is a structure returned by <CODE>hmc('state')</CODE> then
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69 it resets the generator to exactly the same state.
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70
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71 <p>The optional parameters in the <CODE>options</CODE> vector have the following
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72 interpretations.
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73
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74 <p><CODE>options(1)</CODE> is set to 1 to display the energy values and rejection
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75 threshold at each step of the Markov chain. If the value is 2, then the
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76 position vectors at each step are also displayed.
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77
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78 <p><CODE>options(5)</CODE> is set to 1 if momentum persistence is used; default 0, for
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79 complete replacement of momentum variables.
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80
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81 <p><CODE>options(7)</CODE> defines the trajectory length (i.e. the number of leap-frog
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82 steps at each iteration). Minimum value 1.
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83
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84 <p><CODE>options(9)</CODE> is set to 1 to check the user defined gradient function.
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85
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86 <p><CODE>options(14)</CODE> is the number of samples retained from the Markov chain;
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87 default 100.
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88
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89 <p><CODE>options(15)</CODE> is the number of samples omitted from the start of the
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90 chain; default 0.
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91
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92 <p><CODE>options(17)</CODE> defines the momentum used when a persistent update of
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93 (leap-frog) momentum is used. This is bounded to the interval [0, 1).
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94
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95 <p><CODE>options(18)</CODE> is the step size used in leap-frogs; default 1/trajectory
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96 length.
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97
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98 <p><h2>
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99 Examples
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100 </h2>
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101 The following code fragment samples from the posterior distribution of
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102 weights for a neural network.
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103 <PRE>
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104
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105 w = mlppak(net);
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106 [samples, energies] = hmc('neterr', w, options, 'netgrad', net, x, t);
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107 </PRE>
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108
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109
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110 <p><h2>
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111 Algorithm
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112 </h2>
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113
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114 The algroithm follows the procedure outlined in Radford Neal's technical
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115 report CRG-TR-93-1 from the University of Toronto. The stochastic update of
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116 momenta samples from a zero mean unit covariance gaussian.
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117
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118 <p><h2>
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119 See Also
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120 </h2>
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121 <CODE><a href="metrop.htm">metrop</a></CODE><hr>
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122 <b>Pages:</b>
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123 <a href="index.htm">Index</a>
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124 <hr>
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125 <p>Copyright (c) Ian T Nabney (1996-9)
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126
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127
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128 </body>
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129 </html> |
