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In machine learning, particularly in recurrent neural networks (RNNs) and some other sequential models, the term "hidden state" refers to the internal representation of the model at a given time step. The hidden state is a vector or a set of values that summarizes the information learned from the input data up to that point in the sequence. 

For instance, in an RNN, the hidden state is updated at each time step based on the current input and the previous hidden state. The hidden state essentially captures information about the context or dependencies within the sequence. It serves as a kind of memory that allows the model to maintain information about what it has seen so far. Mathematically, if h_t represents the hidden state at time step t, x_t represents the input at time step t, and h_t−1 represents the hidden state from the previous time step, the update equation for the hidden state in a simple RNN might look like: 

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where,    

          W_hh is the weight matrix for the hidden state. 

          W_xh is the weight matrix for the input. 

          b_h is the bias term. 

          f is an activation function. 

The hidden state is crucial in capturing temporal dependencies and patterns in sequential data. It allows the model to consider the entire sequence of inputs when making predictions or classifications. In deep learning models with more complex architectures, such as LSTMs (Long Short-Term Memory) or GRUs (Gated Recurrent Units), the concept of hidden state is still present but with more sophisticated mechanisms for handling long-term dependencies and avoiding the vanishing gradient problem. In machine learning models, the hidden state represents internal information that the model uses to make predictions or generate outputs, but it's not directly observable. The hidden state contains information from past observations and helps the model capture patterns and dependencies in the data.

For example, someone coming inside with a wet umbrella can be thought of as analogous to the concept of a hidden state in machine learning. In this real-world scenario: 

1. Observable Data: 

   The fact that the umbrella is wet is observable to you. This is like the output or the result that a machine learning model might produce. 

2. Hidden State: 

   The hidden state, in this case, could be the weather outside. We cannot directly observe the weather, but the wet umbrella is a result or manifestation of the weather. The wet umbrella is a clue or an indirect indicator of the hidden state (rain outside).  

Therefore, we can consider the wet umbrella as a kind of observable evidence (output) that suggests the presence of rain (hidden state) outside, much like how a hidden state in a machine learning model might capture and represent underlying patterns in a sequence of data. 

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