Probabilistic Reasoning. KR Chowdhary, Professor, Department of Computer Science & Engineering, MBM Engineering College, JNV University, Jodhpur,

Transcription

1 robabilistic Reasoning KR Chowdhary, rofessor, Department of Computer Science & Engineering, MM Engineering College, JNV University, Jodhpur,

2 ayes Theorem and Its applications Review of probability ayesian Reasoning ayesian elief Networks

3 Conditionals Defined Conditionals Rearranging nd also ^ ^ ^

4 Joint Semantics Joint probability distribution the equivalent of truth tables in logic Given a complete truth table you can answer any question you want Given the joint probability distribution over N variables you can answer any question you might want to that involve those variables

5 Joint Semantics With logic you do not need the truth table; you can use inference methods and compositional semantics i.e if I know the truth values for and, I can retrieve the value of ^ With probability, you need the joint to do inference unless you are willing to make some assumptions

6 Joint robabilities Malaria True Malaria False Headache=True Headache=False What s the probability of having a malaria and a headache? What s the probability of having a headache? What s the probability of not having a malaria? What s the probability of having a headache or a malaria?

7 Joint robabilities What about Malaria Headache Do not use ayes theorem MalariaHeadache= Malaria ^ Headache Headache consider only those who have headache

8 Details Where do all the numbers come from? Mostly counting Sometimes theory Sometimes guessing Sometimes all of the above

9 Numbers X Count ll Xs Count ll Events X^Y Count ll X and Y together Count llevents XY Count ll X and Y Together Count ll Ys

10 ayes theorem: We know So rearranging things and

11 ayes theorem: Memorize this:

12 pplications of ayesian Reasoning Diagnosis: Find a likely cause given an observed set of symptoms Categorization: Find a likely category given an observed set of features Hidden rocess Reasoning: Infer characteristics of a hidden process from an observed sequence of outputs from that process

13 pplications In each case we will see that what we want to compute is XY and because of circumstances we have YX; if we can get Y and maybe X we can use ayes theorem. For diagnosis we want CauseSymptoms For categorization: CategoryFeatures

14 ayesian Diagnosis Given a set of symptoms choose the best disease the disease most likely to give rise to those symptoms I.e. Choose the disease that gives the highest DiseaseSymptoms for all possible diseases ut you probably can t assess that So Maximize this Disease Symptoms Symptoms Disease Disease Symptoms

15 ronchitis: Suppose statistics show that following probabilities hold: = ronchitis, C = Cough C C so C C 0.8 * C

16 In general Cause Effect,... Effecti i Effect i i Cause * Cause Effecti

17 ayes Theorem works correctly on the following assumptions: The symptom may only depend on the diagnosis and independent of each other Completeness of the diagnosis Mutual exclusion of the diagnosis single fault assumption Correct and complete statistics of deriving the prior probabilities and conditional symptom-diagnosis probabilities

18 ayesian elief Networks ayesian elief network by earl 1988 offers a computational model for reasoning on a set of data expressed with a causal relationship Dependence of a node is justified on its parent node in a belief network Links between the nodes represent the conditional probabilities for causal influence Causal influence reasoning is not circular

19 ayesian elief Networks: Example for Engine efficiency EX. 1: Worn piston rings W causes excessive oil consumption O, which in turn causes low oil level reading L. Ex. 2: worn piston rings W can cause both blue exhaust as well as low oil level L. In this case we do not know if and L are related. Ex. 3: Low oil level L can be caused either by excessive oil consumption C or by an oil leak E. Then given L, its possible causes may be correlated.

20 ayesian elief Networks examples.. Ex.1 W O L Conditional probability from W to O and O to L is: W, O, L = W * OW * LW, O Ex. 2 W L Joint probability distribution: W,, L = W * W * LW

21 ayesian elief Networks examples.. Ex. 3 C E L Joint probability distribution: C, E, L = C * E * LC,E

22 Steps for creating belief networks 1. For all nodes in the belief network make all links as per cause effect 2. Compute probabilities at the consequent nodes based on the probabilities at the premise nodes