25. Determine Probabilities

To implement these models in code, we need a function to which we can pass model parameters/values and return a probability. Fortunately, we can use a normalized probability density function (PDF). Let's revisit Sebastian's discussion of this topic.

Gaussian Intro

We have implemented this Gaussian Distribution as a C++ function, normpdf , and will practice using it at the end of this concept. normpdf accepts a value, a parameter, and a standard deviation, returning a probability.

Additional Resources for Gaussian Distributions

Let's practice using normpdf to determine transition model probabilities. Specifically, we need to determine the probability of moving from x_{t-1} --> x_t . The value entered into normpdf will be the distance between these two positions. We will refer to potential values of these positions as pseudo position and pre-pseudo position. For example, if our pseudo position x is 8 and our pre-pseudo position is 5, our sample value will be 3, and our transition will be from x - 3 --> x.

To calculate our transition model probability, pass any difference in distances into normpdf using our control parameter and position standard deviation.

Maximize Gaussian

Given pseudo position x and a control parameter of 1 (move 1 unit each time step), which pre-pseudo position maximizes our probability?

SOLUTION: x-1

Start Quiz: 

#include <iostream>
#include "help_functions.h"

// TODO: assign a value, the difference in distances between x_t and x_{t-1}
// that maximizes the returned probability of norm_pdf

float value = ?;  // YOUR VALUE HERE


float parameter = 1.0;  // set as control parameter or observation measurement
float stdev = 1.0;  // position or observation standard deviation

int main() {

  float prob = Helpers::normpdf(value, parameter, stdev);

  std::cout << prob << std::endl;

  return 0;
}
#ifndef HELP_FUNCTIONS_H
#define HELP_FUNCTIONS_H

#include <math.h>

class Helpers {
 public:
  // definition of one over square root of 2*pi:
  constexpr static float STATIC_ONE_OVER_SQRT_2PI = 1/sqrt(2*M_PI);

  /**
   * normpdf(X,mu,sigma) computes the probability function at values x using the
   * normal distribution with mean mu and standard deviation std. x, mu and 
   * sigma must be scalar! The parameter std must be positive. 
   * The normal pdf is y=f(x,mu,std)= 1/(std*sqrt(2pi)) e[ -(x−mu)^2 / 2*std^2 ]
   */
  static float normpdf(float x, float mu, float std) {
    return (STATIC_ONE_OVER_SQRT_2PI/std)*exp(-0.5*pow((x-mu)/std,2));
  }
};

#endif  // HELP_FUNCTIONS_H

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