Since we don't use unknown word models for Naive Bayes, the word 'with' should be dropped, and the remaining words are:  “predictable”, “no”, and “fun”. 
For sentence S = “predictable with no fun”, the computations of probabilities are as follows. The model thus predicts the class negative for the test sentence.

Example of Naive Bayes: Thinking Process

We use a sentiment analysis domain with the two classes positive
(+) and negative (-), and take the following miniature training and test documents simplified from actual movie reviews.

Hamilton College

Naive Bayes Classifier is a probabilistic model. It implements the idea of Bayes Inference and works on two assumptions: Bag-of-Word Assumption and Naive Bayes Assumption.

Navie Bayes Classifier

An on-going but a helpful book resource about NLP
https://web.stanford.edu/~jurafsky/slp3/

Speech and Language Processing (3rd ed. draft) 

Bayes Inference derives the posterior probability based on a prior probability and a likelihood function according to Bayes' Theorem.
$P(C|D)=\frac{P(D|C)P(C)}{P(D)}$
Here, the probability $P(D|C)$ is the likelihood function, and $P(C)$ is the prior probability.

Bayesian Inference

The Naive Bayes classifier assigns the most probable class $$\hat{c}$$ by choosing the class which has the highest probability for a document $$d$$: $$\hat{c}=argmax_{c\in C}P(c|d)$$. Using Bayesian inference, this is equivalent to finding: $$\hat{c}=argmax_{c\in C}P(d|c)P(c)$$, where $$P(d|c)$$ is the likelihood function and $$P(c)$$ is the prior probability.

Bayesian Inference in Naive Bayes Classifier

Naive Bayes Classifier can be viewed as a generative model. Based on:
$\hat{c}=argmax_{c\in C}P(d|c)P(c)$, the generation of a document can be stated as: first, sample a class c from P(c); then, the words in the document can be generated by sampling from P(d|c).

Naive Bayes Classifier as a Generative Model

Bag-of-Word Assumption is used to simplify the computation of Naive Bayes Classifier. It assumes that the positions of words in a document does not affect its classification. Based on this assumption, the features of a document can be simplified as the identical word encoding, discarding their relative ordering. Hence, the likelihood function P(d|c) can be written as $P(f_1,...,f_n|c)$

Bag of Word Assumption

Naive Bayes Assumption is the conditional independence assumption that is used to simplify the computation of the likelihood function $P(f_1,...,f_n|c)=P(f_1|c)...P(f_n|c)$.

Naive Bayes Assumption

The general model of Naive Bayes Classifier can be written as:
$\hat{c}=argmax_{c\in C}\log P(c)+\sum_i \log P(w_i|c)$,
where $C$ is the set of all possible classes, $i$ is the position of word $w_i$ in the text.
This is a linear model, since the prediction is computed as a linear combination of input features, ie., the words.

The General Expression of Naive Bayes Classifier for Text Classification

Sentiment Analysis can be viewed as a classification task with two classes, that is, positive and negative. Naive Bayes Classifier can be modified as Binary Multinomial Naive Bayes Classifier to improve its performance on sentiment analysis.

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