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Anonymous conditional kernels

Definition 30   An anonymous conditional kernel is a conditional kernel where the compute function $f(\Omega_s\vert\Omega_k)$ has been provided by the programmer.

Similarly to the previous section an anonymous conditional kernel is constructed by using an external function that takes a C++ function or class method as parameter, the prototypes are the following:

plProbValue user_function(const plValues &input_values)

or

plProbValue object_class::class_method(const plValues &input_values)

Particularly for an anonymous kernel the parameter input_values contains search and known variables values. Generally speaking and having $n_s$ search variables and $n_k$ known variables the index from $0$ to $(n_s-1)$ on input_values correspond to the search variables; and the index from $n_s$ to $n_s+n_k$ correspond to the known variables.

To follow our previous example here we show the construction of an anonymous conditional kernel representing a family of Laplace distributions, see Figure 4.5.

図 4.5: A family of Laplace distributions with $b=1.118$ and variable mean.

Example 19   Let us suppose that we have a cheap bathroom scale that gives a reading following a Laplace distribution. Let $W_y$ and $W_t$ be the real weight yesterday and today respectively. Let $M_0,M_1,M_2,M_3$ represent four measures taken today on the bathroom scale.The Laplace distribution has a mean ($\mu = W_t$) and a parameter ($b=1.118$). $W_t$ has anormal distribution with mean ($\mu = W_y$) and a variance ($\sigma^2=0.25$). We would like to know the most probable weight today $w_t\in W_t$, knowing the real weight yesterday $w_y\in W_y$ and the four measures. Other wise stated we search the best value of $P(W_t \vert W_y=w_y~M1=m_1~M_2=m_2~M_3=m_3~M_4=m_4)$.

Program 16 implements our example. Lines 22 to 28 corresponds the C++ function coding the conditional Laplace distribution. In this case there is only one search variable and one known variable: x[0] and x[1] respectively. The distribution of yesterdays_weight, a dirac, is constructed at line 55 followed of the construction of the todays_weight a normal distribution. The variable list_of_Pm_tw contains the product of all $P($M(i)$\vert$Wt$)$ that is

$$P(M(0)\vert Wt)*P(M(1)\vert Wt)*P(M(2)\vert Wt)*P(M(3)\vert Wt).$$

Note that each of the $P(M(i)\vert Wt)$ is constructed by using a probability external function and an anonymous kernel, lines 70-73. The first parameter of the external function includes the search and known variables and they must be concatenated in that order. That is, the know variables are concatenated after the search variables. In effect, before calling the C++ function (or method) the external function will arrange the variables values in the same order established at its construction. That is why the order is so important. At the construction of the anonymous conditional kernel the search and known variables are indicated by the first and second parameter measure(i) and todays_weight. Finally, lines 79 to 112 corresponds to the joint distribution and computation of the result.

Program 16: The bathroom scales.

     1  /*=============================================================================
     2   * File           : weight.cpp
     3   *=============================================================================
     4   *
     5   *------------------------- Description ---------------------------------------
     6   * The main goal of this program is to show how to construct a conditional 
     7   * anonymous kernel. The program consists in writing the joint distribution for
     8   * a set of n bathroom scales readings and the yesterday and todays weights.
     9   * The system is asked for the more probable todays weight given the set of
    10   * readings and the yesterdays weight. The reading of the bathroom scales is
    11   * suposed to be a random variable with a laplace distribution centered at the
    12   * real weight.
    13   *-----------------------------------------------------------------------------
    14  */
    15
    16  #include <pl.h>
    17
    18  #define N_MEASURES 4
    19  #define B 1.118
    20  #define WEIGHT_STD 0.25 
    21
    22  plProbValue laplace(const plValues &x)
    23  {
    24    double xx = x[0];
    25    double mu = x[1];
    26
    27    return (1.0/(2.0*B)*exp(-fabs(xx-mu)/B));
    28  }
    29
    30
    31  main()
    32  {
    33    int i;
    34    double ith_measure,y_weight;
    35
    36    /**********************************************************************
    37       Defining the variable type, set and values 
    38    ***********************************************************************/
    39
    40    plRealType weight_type(14,150,1500);
    41    plSymbol todays_weight("todays_weight",weight_type);
    42    plSymbol yesterdays_weight("yesterdays_weight",weight_type);
    43    plArray  measure("measure",weight_type,1,N_MEASURES);
    44    plValues values(todays_weight^yesterdays_weight^measure);
    45
    46    /**********************************************************************
    47      Creating the parametric variables distributions and the 
    48      joint distribution 
    49    ***********************************************************************/
    50
    51    cout<<"Give me your yesterdays weight : ";
    52    cin>>y_weight;
    53    values[yesterdays_weight] = y_weight;
    54    
    55    //P(yesterdays_weight) is a dirac 
    56    plDirac Pyw(yesterdays_weight,values);
    57
    58    // P(todays_weight) = Normal(todays_weight,weight,WEIGHT_STD);
    59    plNormal Ptw(todays_weight,y_weight,WEIGHT_STD);
    60
    61    // Pm_tw is a vector containing all the P(measure_i | todays_weight)
    62    plCndAnonymousKernel *Pm_tw[N_MEASURES];
    63
    64    // list_of_Pm_tw is the product of all P(measure_i | todays_weight)
    65    plComputableObjectList list_of_Pm_tw;
    66    for(i=0;i<N_MEASURES;i++)
    67      {
    68        // Each of the P(measure_i | todays_weight) is a Laplace distribution
    69        // on measure_i with mean equal to todays_weight
    70        plExternalProbFunction laplace_distrib(measure(i)^todays_weight,
    71                                               laplace);
    72        Pm_tw[i] = new plCndAnonymousKernel(measure(i),
    73                                            todays_weight,laplace_distrib);
    74
    75        // Add the new distribution to the list
    76        list_of_Pm_tw *= *Pm_tw[i];
    77      }
    78
    79    // Construction of the joint distribution
    80    plJointDistribution jd(todays_weight^yesterdays_weight^measure,
    81                           Pyw*Ptw*list_of_Pm_tw);
    82
    83    /**********************************************************************
    84      Reading the measures 
    85     ***********************************************************************/
    86
    87    for(i=0;i<N_MEASURES;i++)
    88      {
    89        cout<<"Measure "<<i+1<<"th ? : ";
    90        cin>>ith_measure;
    91        values[measure(i)] = ith_measure;
    92      }
    93
    94    /**********************************************************************
    95      Making the question and showing the result
    96    ***********************************************************************/
    97
    98    // Getting the conditional distribution 
    99    // P(todays_weight | yesterdays_weight measure)
   100    plCndKernel cnd_question;
   101    jd.ask(cnd_question,todays_weight,yesterdays_weight^measure);
   102
   103    // Getting the distribution 
   104    // P(todays_weight | yesterdays_weight=y_weight measure= [m1,m2,m3,...,mn])
   105    plKernel P_todays_weight;
   106    cnd_question.instantiate(P_todays_weight,values);
   107    
   108    // Showing the result
   109    P_todays_weight.best(values);
   110    cout<<"Your more probable weight is : "<<values[todays_weight];
   111    cout<<endl;
   112  }

One output of Program 16 shows as follows;

Give me your yesterdays weight : 77.2  
Measure 1th ? : 76.00
Measure 2th ? : 78.98
Measure 3th ? : 78.27
Measure 4th ? : 78.51

Your more probable weight is : 77.3118.

In this example you can perceive the influence of $w_y$. In fact, if we limit our model to the information given by the bathroom scale (we do not use that of $W_y$), then the result is completely different. For instance, if we change line 59, the distribution of $W_y$, by an uniform distribution

plCUniform Ptw(Wy);

then the result becomes 78.4273.