PSense

Tutorial 1: Modeling Conditional Distribution

Build Models in PSI

PSI is a solver that does exact probabilistic inference. Here is a conditional probabilistic distribution modeled with the source language of PSI.

def main(){
    A:=flip(0.5);
    B:=flip(0.5);
    C:=flip(0.5);
    D:=A+B+C;
    observe(D>=2);
    return A;
}

Here is what the program does:

• The first three lines in the main function simulate the process of tossing three coins (encoded as 1 for head and 0 for tail).

  A:=flip(0.5);
  B:=flip(0.5);
  C:=flip(0.5);
  

  flip(0.5) samples from a Bernoulli distribution with 0.5 probability to be 1 or 0. A, B, and C are three independent samples each with the prior distribution of flip(0.5). Currently the parameter 0.5 is a constant value; different values of the parameter will give different distributions.

• Adding the sampled numbers from the Bernoulli distribution:

  D:=A+B+C;
  

• Suppose we have observed the sum D is greater or equal to 2 (which means at least two coints are heads), we can add the condition as:

  observe(D>=2);
  

• Under the condition that at least two heads are observed, we want to know what is the posterior distribution of A, so:

  return A;
  

• The result given by PSI in Wolfram Language should be:

  p[A_] := (1/2*Boole[-A+1<=0]+1/4*Boole[-A+2<=0]+1/4*Boole[-A<=0])*(DiracDelta[-A+1]+DiracDelta[A])
  

  which is the probability density function of the posterior distribution of A.

The code snippet can be found here.

Find Sensitivity

Sensitivity Definition

The result of probabilistic inference depends on the distribution parameters. In our example above, the prior flip(0.5)(Bernoulli distribution with $p = 0.5$) depends on the constant parameter 0.5. We are interested in the question: what happens to the distribution output if we perturb the parameter of the prior distibution? Notice that our example has 3 constant parameters:

A:=flip(0.5);
B:=flip(0.5);
C:=flip(0.5);

To estimate the change in the output distribution, we can add disturbance ?eps to each of our prior flip(0.5). Conceptually, we may have:

A:=flip(0.5+?eps);
B:=flip(0.5);      //Or, B:=flip(0.5+?eps2);
C:=flip(0.5);      //    C:=flip(0.5+?eps3);

PSense for Sensitivity Analysis

To automatically find the sensitivity of this probabilistic program, run the following in shell prompt:

psense -f examples/conditioning.psi

PSense changes each parameter and gets the results for different metrics:

• Expectation Distance

  It is defined as $D_{Exp}=|\mathbb{E}[p_{eps}(r)]-\mathbb{E}[p(r)]|$, where $\mathbb{E}[p_{eps}(r)]$ and $\mathbb{E}[p(r)]$ are expectations of the output distributions with and without disturbance. After changing the first parameter, PSense genterates the symbolic expression for Expectation Distance as:

  ┌Analyzed parameter 1──────────────────────────────────────────┐
  │            │ Expectation Distance                            │
  ├────────────┼─────────────────────────────────────────────────┤
  │ Expression │ (3*Abs[eps])/(4*(1 + eps))                      │
  └────────────┴─────────────────────────────────────────────────┘
  

  PSense finds the maximum value of the Expectation Distance with respect to the disturbance eps within $\pm 10\%$ of the original parameter:

  ┌Analyzed parameter 1──────────────────────────────────────────┐
  │            │ Expectation Distance                            │
  ├────────────┼─────────────────────────────────────────────────┤
  │ ...        │ ...                                             │
  │ Maximum    │ 0.0394736842105 eps -> -0.0500000000000 r1 -> 0 │
  └────────────┴─────────────────────────────────────────────────┘
  

  The result above shows that the maximum value 0.0395 is obtained when eps $= -0.05$. PSense further analyzes whether the distance grows linearly when the disturbance |eps| increases:

  ┌Analyzed parameter 1──────────────────────────────────────────┐
  │            │ Expectation Distance                            │
  ├────────────┼─────────────────────────────────────────────────┤
  │ ...        │ ...                                             │
  │ Linear     │ False                                           │
  └────────────┴─────────────────────────────────────────────────┘
  

  In this example the Expectation Distance does not grow linearly so PSense outputs False.

• Kolmogorov–Smirnov Statistic

  It is defined as $D_{KS}=\sup_{r\in support}|p_{eps}(r)-p(r)|$, where p_{eps}(r) and p(r) are cumulative density functions, and $\sup_{r\in support}$ represents the supremum of the distance over the support of $r$. PSense gives the distance $|p_{eps}(r)-p(r)|$ and the maximum value of $D_{KS}$, and then analyzes the linearity of $D_{KS}$.

  ┌Analyzed parameter 1───────────────────────────────────────────┐
  │            │ KS Distance                                      │
  ├────────────┼──────────────────────────────────────────────────┤
  │ Expression │ (3*Abs[(eps*(-1 + Boole[r1 >= 1]))/(1 + eps)])/4 │
  │ Maximum    │ 0.0394736842105 eps -> -0.0500000000000 r1 -> 0  │
  │ Linear     │ False                                            │
  └────────────┴──────────────────────────────────────────────────┘
  

• Total Variation Distance

  It is defined as $D_{TVD}=\frac{1}{2}\Sigma_{r\in support}|p_{eps}(r)-p(r)|$ for discrete distributions and $D_{TVD}=\frac{1}{2}\int_{r\in support}|p_{eps}(r)-p(r)|$ for continuous ones. PSense outputs:

  Discrete
  ┌Analyzed parameter 1──────────────────────────────────────────┐
  │            │ TVD                                             │
  ├────────────┼─────────────────────────────────────────────────┤
  │ Expression │ (3*Abs[eps/(1 + eps)])/8                        │
  │ Maximum    │ 0.0197368421053 eps -> -0.0500000000000 r1 -> 0 │
  │ Linear     │ False                                           │
  └────────────┴─────────────────────────────────────────────────┘
  

• Kullback–Leibler Divergence

  Defined as $D_{KL}=\Sigma_{r\in support} p_{eps}(r) \log \frac{p_{eps}(r)}{p(r)}$ for discrete distributions and $D_{KL}=\int_{r\in support} p_{eps}(r) \log \frac{p_{eps}(r)}{p(r)}$ for continuous ones. For this example, it outputs:

  ┌Analyzed parameter 1──────────────────────────────────────────┐
  │            │ KL                                              │
  ├────────────┼─────────────────────────────────────────────────┤
  │ Expression │ ((1 - 2*eps)*Log[-2 + 3/(1 + eps)])/(4*(1 +     │
  │            │ eps)*Log[2])                                    │
  │ Maximum    │ 0.0612248725508 eps -> -0.0499999999962 r1 -> 0 │
  │ Linear     │ False                                           │
  └────────────┴─────────────────────────────────────────────────┘
  

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